title

Lecture 2: Story Proofs, Axioms of Probability | Statistics 110

description

We fill in the "Bose-Einstein" entry of the sampling table, and discuss story proofs. For example, proving Vandermonde's identity with a story is easier and more insightful than going through a tedious algebraic derivation. We then introduce the axioms of probability.

detail

{'title': 'Lecture 2: Story Proofs, Axioms of Probability | Statistics 110', 'heatmap': [{'end': 630.207, 'start': 548.118, 'weight': 0.798}, {'end': 1154.909, 'start': 1089.896, 'weight': 0.785}, {'end': 1593.044, 'start': 1527.952, 'weight': 0.715}, {'end': 2091.899, 'start': 2022.811, 'weight': 0.833}], 'summary': 'The lecture covers various topics including homework and calculations, problem-solving tips, teams splitting and probability homework, combinatorics, its significance in physics and mathematics, and probability foundations with emphasis on non-naive definition, probability space, and axioms of probability.', 'chapters': [{'end': 190.122, 'segs': [{'end': 73.309, 'src': 'embed', 'start': 0.089, 'weight': 0, 'content': [{'end': 9.275, 'text': 'Okay, so as far as clarifications and hints and comments and so on, on the homework, these are kind of general comments.', 'start': 0.089, 'duration': 9.186}, {'end': 12.578, 'text': "One is don't lose your common sense.", 'start': 9.936, 'duration': 2.642}, {'end': 21.861, 'text': "That doesn't mean you can rely only on common sense,", 'start': 19.298, 'duration': 2.563}, {'end': 28.249, 'text': "because we're going to see over and over again in this course a lot of counterintuitive results that may seem to defy common sense.", 'start': 21.861, 'duration': 6.388}, {'end': 33.596, 'text': "But just because we're doing some counterintuitive stuff sometimes doesn't mean you should abandon common sense.", 'start': 28.51, 'duration': 5.086}, {'end': 42.7, 'text': 'And I mean this not only in terms of do your answers make sense, but also just in terms of being reasonable.', 'start': 35.477, 'duration': 7.223}, {'end': 46.602, 'text': 'Like a couple of you asked about calculators and things like that.', 'start': 43.081, 'duration': 3.521}, {'end': 51.944, 'text': "On the homework, you can use calculators if you want, but it shouldn't be necessary for the most part.", 'start': 48.003, 'duration': 3.941}, {'end': 56.266, 'text': "Once in a while, it'll be obvious that you should use a calculator for that part.", 'start': 52.084, 'duration': 4.182}, {'end': 60.288, 'text': "On the exams, there's no calculators allowed, but also no calculators needed.", 'start': 56.766, 'duration': 3.522}, {'end': 65.167, 'text': 'So either on the homework or on the exam, you should use some common sense.', 'start': 61.246, 'duration': 3.921}, {'end': 70.988, 'text': "For example, if you have 52 choose 5, it's perfectly fine to leave it as 52 choose 5.", 'start': 65.547, 'duration': 5.441}, {'end': 73.309, 'text': 'Certainly on the exam, common sense.', 'start': 70.988, 'duration': 2.321}], 'summary': 'Use common sense in homework and exams. calculators allowed for homework, not needed. no calculators allowed in exams.', 'duration': 73.22, 'max_score': 0.089, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg89.jpg'}, {'end': 190.122, 'src': 'embed', 'start': 96.654, 'weight': 2, 'content': [{'end': 101.88, 'text': 'But 52 choose 5 is already making you think, oh, this has something to do with choosing 5 out of 52.', 'start': 96.654, 'duration': 5.226}, {'end': 103.742, 'text': 'So this is self-annotating, which is good.', 'start': 101.88, 'duration': 1.862}, {'end': 106.245, 'text': 'You could leave it that way.', 'start': 105.284, 'duration': 0.961}, {'end': 109.989, 'text': 'On the other hand, So this you would leave.', 'start': 106.485, 'duration': 3.504}, {'end': 123.379, 'text': 'But common sense would be, well, if you have 4 divided by 2 times 1, either on the homework or on the exam, I would prefer that you simplify it to 2.', 'start': 110.51, 'duration': 12.869}, {'end': 125.62, 'text': 'Occasionally, you may need to add fractions.', 'start': 123.379, 'duration': 2.241}, {'end': 129.523, 'text': "So 1 half plus 1 third, I'm assuming you can do without a calculator.", 'start': 125.921, 'duration': 3.602}, {'end': 135.346, 'text': "But you're never going to have ugly, tedious stuff on the exam, certainly.", 'start': 129.562, 'duration': 5.784}, {'end': 139.228, 'text': "And if it's tedious on the homework, then you could use a calculator for that.", 'start': 135.426, 'duration': 3.802}, {'end': 141.87, 'text': "And that's also a hint.", 'start': 140.949, 'duration': 0.921}, {'end': 148.774, 'text': 'If, on the midterm or final, you find yourself doing all these massive calculations with huge fractions and things like that,', 'start': 142.21, 'duration': 6.564}, {'end': 150.195, 'text': "then there's probably something wrong.", 'start': 148.774, 'duration': 1.421}, {'end': 154.516, 'text': "OK So that's just a clarification about that.", 'start': 151.696, 'duration': 2.82}, {'end': 160.942, 'text': 'Second, useful throughout the course, do check answers.', 'start': 156.778, 'duration': 4.164}, {'end': 166.536, 'text': "And I'll talk more about checking answers at different points.", 'start': 162.675, 'duration': 3.861}, {'end': 173.697, 'text': 'But just briefly, checking answers does not mean go through and look what you did and say, that looks OK.', 'start': 167.356, 'duration': 6.341}, {'end': 176.198, 'text': 'Do the same thing twice.', 'start': 174.598, 'duration': 1.6}, {'end': 180.839, 'text': "Because if you made a mistake the first time, you're probably not going to detect it when you look through it again.", 'start': 176.978, 'duration': 3.861}, {'end': 186.2, 'text': 'Checking answer means trying special cases, things like that.', 'start': 181.199, 'duration': 5.001}, {'end': 190.122, 'text': 'thinking of another approach, a lot of problems can be solved in more than one method.', 'start': 186.58, 'duration': 3.542}], 'summary': 'Mathematics tips: focus on simplifying, avoid tedious calculations, and use multiple problem-solving methods.', 'duration': 93.468, 'max_score': 96.654, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg96654.jpg'}], 'start': 0.089, 'title': 'Homework and calculations', 'summary': 'Emphasizes common sense and reasonableness in approaching homework, highlights the occurrence of counterintuitive results, and the limited necessity of calculators. it also focuses on using common sense to handle calculations, simplifying expressions, adding fractions, mental calculation, and checking answers.', 'chapters': [{'end': 51.944, 'start': 0.089, 'title': 'Homework clarifications', 'summary': 'Emphasizes the importance of common sense and reasonableness in approaching homework, highlighting the occurrence of counterintuitive results in the course content and the limited necessity of calculators for the homework.', 'duration': 51.855, 'highlights': ['The chapter stresses the significance of common sense and reasonableness in addressing homework, despite the presence of counterintuitive results in the course content.', 'It mentions that calculators are not necessary for the most part when working on the homework.']}, {'end': 123.379, 'start': 52.084, 'title': 'Calculating combinations and simplifying expressions', 'summary': "Emphasizes using common sense to handle calculations, such as leaving complex combinations as is and simplifying expressions when necessary, as it's essential for exams where calculators are not allowed.", 'duration': 71.295, 'highlights': ["The chapter stresses the importance of using common sense for calculations, such as leaving complex combinations as they are, as it's crucial for the exam where calculators are not allowed.", 'It is perfectly fine to leave complex combinations like 52 choose 5 as they are, as they are self-annotating and provide insight into the problem, which is beneficial for understanding the context of the calculation.', 'Simplifying expressions, such as 4 divided by 2 times 1, to 2 is preferred on both homework and exams, showcasing the practical application of common sense in mathematical problem-solving.']}, {'end': 190.122, 'start': 123.379, 'title': 'Adding fractions and checking answers', 'summary': 'Highlights the importance of mental calculation for adding fractions and emphasizes the use of calculators for tedious calculations. it also stresses the significance of checking answers by trying special cases and exploring different problem-solving methods.', 'duration': 66.743, 'highlights': ['The significance of mental calculation over the use of calculators for adding fractions is emphasized, with the assurance that the exam will not have tedious calculations, and calculators can be used for homework. (Quantifiable data: emphasis on mental calculation)', 'The importance of checking answers by trying special cases and employing different problem-solving methods is stressed, as it can reveal mistakes that may not be detected through a simple reevaluation of the initial approach. (Quantifiable data: emphasis on checking answers)', 'The chapter provides a hint that if students find themselves consistently performing massive calculations with huge fractions in the midterm or final, there may be an issue that needs to be addressed. (Quantifiable data: indication of potential problem)']}], 'duration': 190.033, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg89.jpg', 'highlights': ['The chapter emphasizes the significance of common sense and reasonableness in addressing homework, despite the presence of counterintuitive results in the course content.', "The chapter stresses the importance of using common sense for calculations, such as leaving complex combinations as they are, as it's crucial for the exam where calculators are not allowed.", 'Simplifying expressions, such as 4 divided by 2 times 1, to 2 is preferred on both homework and exams, showcasing the practical application of common sense in mathematical problem-solving.', 'The significance of mental calculation over the use of calculators for adding fractions is emphasized, with the assurance that the exam will not have tedious calculations, and calculators can be used for homework.', 'The importance of checking answers by trying special cases and employing different problem-solving methods is stressed, as it can reveal mistakes that may not be detected through a simple reevaluation of the initial approach.', 'The chapter provides a hint that if students find themselves consistently performing massive calculations with huge fractions in the midterm or final, there may be an issue that needs to be addressed.', 'It mentions that calculators are not necessary for the most part when working on the homework.']}, {'end': 465.93, 'segs': [{'end': 246.848, 'src': 'embed', 'start': 214.204, 'weight': 1, 'content': [{'end': 222.471, 'text': "And if you plug in n equals 1, and it's completely obviously wrong then that's useful information, right?", 'start': 214.204, 'duration': 8.267}, {'end': 226.975, 'text': "But most students don't bother to just plug in n equals 1 and see if it makes sense.", 'start': 222.591, 'duration': 4.384}, {'end': 229.137, 'text': 'So simple and extreme cases.', 'start': 227.515, 'duration': 1.622}, {'end': 246.848, 'text': "The third thing, I've never seen it given a name or any emphasis at all, but it's very, very useful to label.", 'start': 235.441, 'duration': 11.407}], 'summary': 'Testing n=1 is useful for identifying errors in mathematical expressions.', 'duration': 32.644, 'max_score': 214.204, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg214204.jpg'}, {'end': 358.753, 'src': 'embed', 'start': 329.437, 'weight': 0, 'content': [{'end': 333, 'text': "It's very useful if you, for example, you can choose your own notation.", 'start': 329.437, 'duration': 3.563}, {'end': 339.984, 'text': "It's very useful, for example, though, you could say, let's number or label the red balls 1 through r and the green balls.", 'start': 333.52, 'duration': 6.464}, {'end': 343.586, 'text': 'R plus 1 through R plus G, for example.', 'start': 340.965, 'duration': 2.621}, {'end': 344.927, 'text': 'You could do something like that.', 'start': 343.606, 'duration': 1.321}, {'end': 349.249, 'text': "Then you're referring to specific balls.", 'start': 345.267, 'duration': 3.982}, {'end': 352.01, 'text': 'Now this might seem very obvious,', 'start': 349.709, 'duration': 2.301}, {'end': 358.753, 'text': "but I'm stressing it just because I've seen students very often get confused because they're not thinking of this labeling.", 'start': 352.01, 'duration': 6.743}], 'summary': 'Choose your own notation for labeling specific balls to avoid confusion.', 'duration': 29.316, 'max_score': 329.437, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg329437.jpg'}, {'end': 416.956, 'src': 'embed', 'start': 376.509, 'weight': 2, 'content': [{'end': 379.551, 'text': 'are they indistinguishable or distinguishable and whatever?', 'start': 376.509, 'duration': 3.042}, {'end': 390.317, 'text': 'So suppose you had, like you know, 10 green balls in a jar and they all look completely identical to you.', 'start': 380.511, 'duration': 9.806}, {'end': 395.06, 'text': "okay?. I'm not actually saying that the numbers are actually written on them.", 'start': 390.317, 'duration': 4.743}, {'end': 397.341, 'text': 'They may look completely identical to you.', 'start': 395.38, 'duration': 1.961}, {'end': 406.387, 'text': 'But the point is that as far as probability is concerned, as far as nature is concerned, it behaves as if they are distinguishable and labeled.', 'start': 397.641, 'duration': 8.746}, {'end': 409.129, 'text': "And that's going to give you the correct answers.", 'start': 406.727, 'duration': 2.402}, {'end': 414.954, 'text': "Whereas if you just say, well, they're completely identical, so they're indistinguishable, you'll run into trouble.", 'start': 409.149, 'duration': 5.805}, {'end': 416.956, 'text': 'So it helps to think about the labeling.', 'start': 415.234, 'duration': 1.722}], 'summary': 'Probability behaves as if indistinguishable objects are distinguishable and labeled, giving correct answers.', 'duration': 40.447, 'max_score': 376.509, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg376509.jpg'}], 'start': 190.142, 'title': 'Problem-solving tips and labeling objects', 'summary': 'Emphasizes the use of simple and extreme cases, along with labeling objects, to aid in problem-solving, demonstrating the value of plugging in n equals 1 and using labels for objects. it also discusses the importance of labeling and distinguishability in probability, illustrated through examples of balls in a jar and robberies in districts.', 'chapters': [{'end': 352.01, 'start': 190.142, 'title': 'Problem-solving tips and labeling objects', 'summary': 'Emphasizes the importance of using simple and extreme cases, along with labeling objects, to aid in problem-solving and better understanding, demonstrating the value of plugging in n equals 1 and using labels for objects.', 'duration': 161.868, 'highlights': ['Using simple and extreme cases, like plugging in n equals 1, helps in obtaining useful information and learning from different answers.', 'Labeling objects, such as people or animals, with numbers from 1 up to n, aids in problem-solving and understanding, especially in scenarios involving collections of items like balls in a jar.']}, {'end': 465.93, 'start': 352.01, 'title': 'Labeling and distinguishability in probability', 'summary': 'Discusses the importance of labeling and distinguishability in probability, emphasizing that even seemingly identical items should be treated as distinguishable to obtain correct answers, illustrated through examples of balls in a jar and robberies in districts.', 'duration': 113.92, 'highlights': ['The importance of treating seemingly identical items as distinguishable in probability to obtain correct answers, as even identical twins can be told apart in some way.', 'Illustration of treating seemingly identical balls in a jar as distinguishable in probability, despite their physical appearance, to ensure accurate probability calculations.', 'Emphasizing the relevance of labeling and distinguishability in understanding the nature of events, as even similar robberies in different districts can be distinguished by their unique circumstances and should be treated as distinguishable for accurate analysis.']}], 'duration': 275.788, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg190142.jpg', 'highlights': ['Labeling objects with numbers from 1 to n aids in problem-solving and understanding.', 'Using simple and extreme cases, like plugging in n equals 1, helps in obtaining useful information and learning from different answers.', 'Emphasizing the relevance of labeling and distinguishability in understanding the nature of events.', 'The importance of treating seemingly identical items as distinguishable in probability to obtain correct answers.', 'Illustration of treating seemingly identical balls in a jar as distinguishable in probability.']}, {'end': 978.089, 'segs': [{'end': 519.982, 'src': 'embed', 'start': 486.894, 'weight': 0, 'content': [{'end': 494.829, 'text': 'If I say you have ten people and you wanna split them into, a team of 4, and a team of 6.', 'start': 486.894, 'duration': 7.935}, {'end': 498.451, 'text': "Let me just do that as a quick example, because there's actually some interesting things here.", 'start': 494.829, 'duration': 3.622}, {'end': 502.453, 'text': "It's a bit of a hint, but it's also useful.", 'start': 499.051, 'duration': 3.402}, {'end': 517.941, 'text': 'So suppose we have 10 people, and I want to split into a team of 6 and a team of 4.', 'start': 502.953, 'duration': 14.988}, {'end': 519.982, 'text': 'And I want to know how many ways are there to do that.', 'start': 517.941, 'duration': 2.041}], 'summary': 'Ten people split into teams of 4 and 6, exploring ways to do so.', 'duration': 33.088, 'max_score': 486.894, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg486894.jpg'}, {'end': 639.32, 'src': 'heatmap', 'start': 548.118, 'weight': 4, 'content': [{'end': 548.919, 'text': 'It must be the same.', 'start': 548.118, 'duration': 0.801}, {'end': 550.62, 'text': 'I mean, you can check this.', 'start': 549.239, 'duration': 1.381}, {'end': 555.584, 'text': "This is 10 factorial over 6 factorial, 4 factorial, and so is this, so it's true.", 'start': 551.401, 'duration': 4.183}, {'end': 558.127, 'text': 'But we proved it just by thinking about it.', 'start': 555.664, 'duration': 2.463}, {'end': 570.887, 'text': "On the other hand, if we wanted two teams of five, Now if we just do 10, choose 5, we're going to be off OK?", 'start': 559.208, 'duration': 11.679}, {'end': 574.79, 'text': 'Because I say pick 5 for one team and the remaining 5 for the other team.', 'start': 570.987, 'duration': 3.803}, {'end': 577.933, 'text': 'But if I had picked the remaining 5 first, that would have been the same.', 'start': 575.191, 'duration': 2.742}, {'end': 583.057, 'text': "Because it's not like I said that there's a team A and a team B that have some difference between them.", 'start': 578.333, 'duration': 4.724}, {'end': 584.378, 'text': "It's just two teams.", 'start': 583.077, 'duration': 1.301}, {'end': 587.961, 'text': 'So picking play, again, it helps to label them.', 'start': 584.418, 'duration': 3.543}, {'end': 590.643, 'text': 'So assume the people are numbered 1 through 5.', 'start': 588.281, 'duration': 2.362}, {'end': 594.886, 'text': "If the teams are 1 through 5 and 6 through 10, there's only one way to do that.", 'start': 590.643, 'duration': 4.243}, {'end': 598.787, 'text': "Right? I'm not designating some distinction between the two teams.", 'start': 595.086, 'duration': 3.701}, {'end': 604.669, 'text': "Therefore, in this case, it would be 10 choose 5 divided by 2 because we've double counted.", 'start': 599.187, 'duration': 5.482}, {'end': 609.651, 'text': "Right? Does everyone understand the difference between these two things? Here, you're dividing by 2.", 'start': 604.689, 'duration': 4.962}, {'end': 610.551, 'text': "Here, you're not.", 'start': 609.651, 'duration': 0.9}, {'end': 615.233, 'text': 'Because there is a clear difference between a team of 4 and a team of 6.', 'start': 611.112, 'duration': 4.121}, {'end': 623.636, 'text': "Right? Well, two teams of 5, unless you said, well, one team is supposed to wear this jersey and the other one wears this jersey, it's equivalent.", 'start': 615.233, 'duration': 8.403}, {'end': 630.207, 'text': "Okay, so that's a key, sometimes subtle distinction that sometimes gets missed, so I wanna emphasize that a little bit.", 'start': 624.935, 'duration': 5.272}, {'end': 632.775, 'text': "All right, so there's a difference there.", 'start': 631.394, 'duration': 1.381}, {'end': 635.417, 'text': 'So you should think carefully about issues like that.', 'start': 632.795, 'duration': 2.622}, {'end': 639.32, 'text': "It's a little too simplistic to say order matters or order doesn't matter.", 'start': 635.798, 'duration': 3.522}], 'summary': 'Discussed combinatorics and the distinction between team selections, highlighting the difference between 10 choose 5 and 10 choose 5 divided by 2.', 'duration': 91.202, 'max_score': 548.118, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg548118.jpg'}, {'end': 693.422, 'src': 'embed', 'start': 659.096, 'weight': 1, 'content': [{'end': 664.579, 'text': "The naive definition assumes you've broken up your problem into equally likely outcomes.", 'start': 659.096, 'duration': 5.483}, {'end': 670.323, 'text': "So if you break up the problem in a way where they're clearly not equally likely, then a naive definition will not work.", 'start': 664.939, 'duration': 5.384}, {'end': 677.947, 'text': "So what I'm saying is for every probability question on the homework, if you frame it in the right way, you can apply the naive definition.", 'start': 670.983, 'duration': 6.964}, {'end': 683.33, 'text': 'But you have to think hard about making sure that it makes sense to assume equally likely outcomes.', 'start': 678.467, 'duration': 4.863}, {'end': 693.422, 'text': 'Does that make sense? Okay, so coming back to this sampling table.', 'start': 683.35, 'duration': 10.072}], 'summary': 'Naive definition applies when problem has equally likely outcomes, requires careful framing of probability questions.', 'duration': 34.326, 'max_score': 659.096, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg659096.jpg'}, {'end': 861.319, 'src': 'embed', 'start': 828.582, 'weight': 2, 'content': [{'end': 832.628, 'text': 'The most extreme example I can think of is k equals 0.', 'start': 828.582, 'duration': 4.046}, {'end': 834.59, 'text': "K equals 0 says you don't do anything.", 'start': 832.628, 'duration': 1.962}, {'end': 838.816, 'text': "So k equals 0, but let's just see if it makes sense.", 'start': 836.373, 'duration': 2.443}, {'end': 844.143, 'text': 'Assuming that this is correct, k equals 0, this is n minus 1 choose 0.', 'start': 839.517, 'duration': 4.626}, {'end': 850.216, 'text': 'n minus 1 choose 0 is 1.', 'start': 844.143, 'duration': 6.073}, {'end': 851.496, 'text': 'For the same reasoning.', 'start': 850.216, 'duration': 1.28}, {'end': 853.917, 'text': 'why 0, factorial 1?', 'start': 851.496, 'duration': 2.421}, {'end': 861.319, 'text': "Well, I always think of factorial in terms of if you have n people and they're lined up for ice cream, how many ways can you order them?", 'start': 853.917, 'duration': 7.402}], 'summary': 'In mathematics, k equals 0 leads to n minus 1 choose 0, resulting in 1, explained using factorial and permutations.', 'duration': 32.737, 'max_score': 828.582, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg828582.jpg'}, {'end': 984.55, 'src': 'embed', 'start': 957.185, 'weight': 3, 'content': [{'end': 961.646, 'text': "You can try n equals 1 is another easy case, but let's do n equals 2.", 'start': 957.185, 'duration': 4.461}, {'end': 965.386, 'text': 'n equals 2 is what I would call the simplest non-trivial example.', 'start': 961.646, 'duration': 3.74}, {'end': 972.848, 'text': "And that's one of the best just general pieces of research advice in general is to look at the simplest non-trivial example.", 'start': 966.047, 'duration': 6.801}, {'end': 976.468, 'text': 'So this one is special in some sense.', 'start': 974.368, 'duration': 2.1}, {'end': 978.089, 'text': 'Simplest non-trivial.', 'start': 976.889, 'duration': 1.2}, {'end': 982.61, 'text': "These ones are pretty trivial, but they're still worth checking, because if they were wrong, then we know there's something wrong.", 'start': 978.149, 'duration': 4.461}, {'end': 984.55, 'text': 'This is the simplest non-trivial.', 'start': 983.03, 'duration': 1.52}], 'summary': 'The research advice is to look at simplest non-trivial examples, like n equals 2, to ensure correctness.', 'duration': 27.365, 'max_score': 957.185, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg957185.jpg'}], 'start': 465.97, 'title': 'Teams splitting and probability homework', 'summary': 'Discusses methods of splitting groups into teams, differentiating between indistinguishable and distinguishable teams, and applying combinations. it also covers the application of the naive definition of probability, use of sampling tables, and derivation of the formula n plus k minus 1 choose k for selecting k times from a set of n objects.', 'chapters': [{'end': 635.417, 'start': 465.97, 'title': 'Teams splitting and counting', 'summary': 'Discusses the different ways of splitting a group into teams and emphasizes the distinction between indistinguishable and distinguishable teams, with examples of applying combinations to calculate the number of ways to split the group.', 'duration': 169.447, 'highlights': ['The chapter explains the process of splitting a group of 10 people into a team of 4 and a team of 6 using combinations, demonstrating that 10 choose 4 equals 10 choose 6, highlighting the concept of indistinguishability and providing a clear example.', 'It emphasizes the distinction between indistinguishable and distinguishable teams, illustrating the difference between splitting into two teams of 5 and two teams of 5 with jerseys, and clarifies the need to carefully consider such distinctions.', 'The discussion touches on the topic of double counting when splitting into two teams of 5, pointing out that in this case, the number of ways is divided by 2 due to the lack of distinction between the teams.']}, {'end': 978.089, 'start': 635.798, 'title': 'Probability homework and sampling table', 'summary': 'Discusses the application of the naive definition of probability to probability questions, the use of the sampling table to determine outcomes, and the derivation of the formula n plus k minus 1 choose k for selecting k times from a set of n objects with replacement and order not mattering.', 'duration': 342.291, 'highlights': ['The chapter discusses the application of the naive definition of probability to probability questions, emphasizing the importance of framing the problem in a way that assumes equally likely outcomes, and mentions that the naive definition is sufficient for the probability questions in the homework.', 'The sampling table is mentioned, with three of the four entries being filled in using the multiplication rule, and the focus on explaining the derivation and intuition behind the tricky fourth entry formula n plus k minus 1 choose k for selecting k times from a set of n objects with replacement and order not mattering.', 'The explanation and validation of the formula n plus k minus 1 choose k is provided through simple and extreme cases, such as k equals 0 resulting in n minus 1 choose 0 equalling 1, and k equals 1 resulting in n choose 1 equalling n, showing the plausibility and correctness of the formula.', 'The chapter also emphasizes the importance of looking at the simplest non-trivial example, illustrated through the case of n equals 2, as a valuable research approach.']}], 'duration': 512.119, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg465970.jpg', 'highlights': ['The chapter explains the process of splitting a group of 10 people into a team of 4 and a team of 6 using combinations, demonstrating that 10 choose 4 equals 10 choose 6, highlighting the concept of indistinguishability and providing a clear example.', 'The chapter discusses the application of the naive definition of probability to probability questions, emphasizing the importance of framing the problem in a way that assumes equally likely outcomes, and mentions that the naive definition is sufficient for the probability questions in the homework.', 'The explanation and validation of the formula n plus k minus 1 choose k is provided through simple and extreme cases, such as k equals 0 resulting in n minus 1 choose 0 equalling 1, and k equals 1 resulting in n choose 1 equalling n, showing the plausibility and correctness of the formula.', 'The chapter also emphasizes the importance of looking at the simplest non-trivial example, illustrated through the case of n equals 2, as a valuable research approach.', 'It emphasizes the distinction between indistinguishable and distinguishable teams, illustrating the difference between splitting into two teams of 5 and two teams of 5 with jerseys, and clarifies the need to carefully consider such distinctions.']}, {'end': 1387.315, 'segs': [{'end': 1065.82, 'src': 'embed', 'start': 1008.129, 'weight': 0, 'content': [{'end': 1014.012, 'text': "Same argument, why is 10 choose 4 the same as 10 choose 6? You could choose the 4 or you could choose the 6, they're the same thing.", 'start': 1008.129, 'duration': 5.883}, {'end': 1016.993, 'text': "This is exactly the same reason, so that's the same as that.", 'start': 1014.452, 'duration': 2.541}, {'end': 1022.636, 'text': "K plus 1 choose 1 is obviously K plus 1, cuz you're choosing one thing out of K plus 1.", 'start': 1017.274, 'duration': 5.362}, {'end': 1024.237, 'text': "All right, now let's see if that's correct.", 'start': 1022.636, 'duration': 1.601}, {'end': 1035.566, 'text': 'We have n, we have two objects, okay, and We are picking k times.', 'start': 1026.778, 'duration': 8.788}, {'end': 1041.957, 'text': "So let's just draw two buckets to represent the two objects.", 'start': 1036.347, 'duration': 5.61}, {'end': 1047.424, 'text': 'k times.', 'start': 1046.363, 'duration': 1.061}, {'end': 1049.005, 'text': "so let's just put a.", 'start': 1047.424, 'duration': 1.581}, {'end': 1051.948, 'text': "you can put a check mark, but for simplicity, I'll put a dot.", 'start': 1049.005, 'duration': 2.943}, {'end': 1053.949, 'text': 'every time I put this is object one.', 'start': 1051.948, 'duration': 2.001}, {'end': 1054.85, 'text': 'this is object two.', 'start': 1053.949, 'duration': 0.901}, {'end': 1058.974, 'text': "okay?, Every time I select object one, I'm gonna put a dot here.", 'start': 1054.85, 'duration': 4.124}, {'end': 1065.82, 'text': "Every time I select object two, I'm gonna put a dot here, okay? So, well, you can make up some numbers if you want.", 'start': 1058.994, 'duration': 6.826}], 'summary': 'Explaining the concept of choosing objects, with an example of k+1 choose 1 resulting in k+1.', 'duration': 57.691, 'max_score': 1008.129, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1008129.jpg'}, {'end': 1177.047, 'src': 'heatmap', 'start': 1089.896, 'weight': 1, 'content': [{'end': 1097.443, 'text': 'okay?. It follows from that that, in order to specify this result,', 'start': 1089.896, 'duration': 7.547}, {'end': 1106.228, 'text': 'all we need to do is say how many dots are there in this box? Because if I know how many dots are in this box,', 'start': 1097.443, 'duration': 8.785}, {'end': 1108.189, 'text': 'then however many are left are in this one.', 'start': 1106.228, 'duration': 1.961}, {'end': 1112.492, 'text': 'So the number of dots here, number of dots.', 'start': 1109.21, 'duration': 3.282}, {'end': 1122.204, 'text': 'is in the set 0, 1, up to k, right? Because I have k dots.', 'start': 1114.439, 'duration': 7.765}, {'end': 1126.066, 'text': 'In my example, k is 7, but k could be whatever, k dots.', 'start': 1122.764, 'duration': 3.302}, {'end': 1132.91, 'text': "So however many are in here, it could be that they're all in here, or none of them, or anything in between.", 'start': 1128.568, 'duration': 4.342}, {'end': 1135.132, 'text': 'So there are k plus 1 possibilities.', 'start': 1133.231, 'duration': 1.901}, {'end': 1142.463, 'text': 'So just by thinking of this, two boxes, that gives a direct proof that this is correct in this case.', 'start': 1135.62, 'duration': 6.843}, {'end': 1146.685, 'text': 'OK, so that gives some comfort.', 'start': 1143.183, 'duration': 3.502}, {'end': 1150.507, 'text': "And I'll come back a little bit to this case, n equals 2.", 'start': 1147.565, 'duration': 2.942}, {'end': 1154.909, 'text': "But now let's prove that this formula is correct in general.", 'start': 1150.507, 'duration': 4.402}, {'end': 1156.67, 'text': 'All right.', 'start': 1156.209, 'duration': 0.461}, {'end': 1168.919, 'text': 'well, this dot picture is already giving us a hint that the problem, One of the most difficult and most important things in this course,', 'start': 1156.67, 'duration': 12.249}, {'end': 1173.403, 'text': 'is to try to get in the habit of trying to recognize pattern and structure.', 'start': 1168.919, 'duration': 4.484}, {'end': 1177.047, 'text': 'That is, recognizing when two problems are equivalent, even if they sound different.', 'start': 1173.464, 'duration': 3.583}], 'summary': 'The number of dots in two boxes is represented by k, with k+1 possibilities and n equals 2.', 'duration': 48.479, 'max_score': 1089.896, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1089896.jpg'}, {'end': 1269.193, 'src': 'embed', 'start': 1229.728, 'weight': 3, 'content': [{'end': 1233.23, 'text': 'The boxes are distinguishable because this is box one and this is box two.', 'start': 1229.728, 'duration': 3.502}, {'end': 1246.697, 'text': 'K indistinguishable particles into n distinguishable boxes.', 'start': 1234.971, 'duration': 11.726}, {'end': 1256.482, 'text': "OK, and the answer is n plus k minus 1 choose k, but let's see why.", 'start': 1252.839, 'duration': 3.643}, {'end': 1264.689, 'text': 'So another piece of advice that I could have added there is draw a diagram and try some simple examples.', 'start': 1256.502, 'duration': 8.187}, {'end': 1269.193, 'text': "I mentioned the simple examples, but I didn't mention draw a diagram.", 'start': 1266.53, 'duration': 2.663}], 'summary': 'Formula for placing k indistinguishable particles into n distinguishable boxes is n + k - 1 choose k.', 'duration': 39.465, 'max_score': 1229.728, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1229728.jpg'}], 'start': 978.149, 'title': 'Combinatorics and counting particles', 'summary': 'Discusses combinatorics example with n=2, showing k+1 choose k equals k+1 and providing a visual representation with 2 objects and k selections. it also explains counting indistinguishable particles in distinguishable boxes using the formula n+k-1 choose k, emphasizing recognizing patterns and structure in problem-solving.', 'chapters': [{'end': 1150.507, 'start': 978.149, 'title': 'Combinatorics example with n=2', 'summary': 'Discusses the combinatorics example with n=2, demonstrating how k plus 1 choose k equals k plus 1, and providing a visual representation with two objects and k selections, leading to k plus 1 possibilities.', 'duration': 172.358, 'highlights': ['Demonstrating k plus 1 choose k equals k plus 1 The explanation and visual representation illustrate the relationship between k plus 1 choose k and k plus 1, providing a clear understanding of the concept.', 'Visual representation with two objects and k selections The visual representation using two objects and k selections provides a tangible and comprehensible example of the concept.', 'Providing a direct proof with k plus 1 possibilities The direct proof using the visual representation establishes the correctness of k plus 1 choose k equals k plus 1, showcasing k plus 1 possibilities in the given scenario.']}, {'end': 1387.315, 'start': 1150.507, 'title': 'Counting particles in boxes', 'summary': 'Explains the concept of counting indistinguishable particles in distinguishable boxes, providing the formula n+k-1 choose k, and emphasizes the importance of recognizing patterns and structure in problem-solving.', 'duration': 236.808, 'highlights': ['The chapter explains the concept of counting indistinguishable particles in distinguishable boxes, providing the formula n+k-1 choose k. The answer to putting k indistinguishable particles into n distinguishable boxes is n+k-1 choose k.', 'The chapter emphasizes the importance of recognizing patterns and structure in problem-solving. One of the most important things in the course is to recognize pattern and structure in problems.']}], 'duration': 409.166, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg978149.jpg', 'highlights': ['Demonstrating k plus 1 choose k equals k plus 1 The explanation and visual representation illustrate the relationship between k plus 1 choose k and k plus 1, providing a clear understanding of the concept.', 'Providing a direct proof with k plus 1 possibilities The direct proof using the visual representation establishes the correctness of k plus 1 choose k equals k plus 1, showcasing k plus 1 possibilities in the given scenario.', 'Visual representation with two objects and k selections The visual representation using two objects and k selections provides a tangible and comprehensible example of the concept.', 'The chapter explains the concept of counting indistinguishable particles in distinguishable boxes, providing the formula n+k-1 choose k. The answer to putting k indistinguishable particles into n distinguishable boxes is n+k-1 choose k.', 'The chapter emphasizes the importance of recognizing patterns and structure in problem-solving. One of the most important things in the course is to recognize pattern and structure in problems.']}, {'end': 2330.186, 'segs': [{'end': 1423.157, 'src': 'embed', 'start': 1387.355, 'weight': 1, 'content': [{'end': 1388.676, 'text': "That's the most common scenario.", 'start': 1387.355, 'duration': 1.321}, {'end': 1393.701, 'text': 'For some physics problems, it behaves more like this, and also for some counting problems.', 'start': 1389.076, 'duration': 4.625}, {'end': 1398.663, 'text': "I'm just emphasizing that this is important in physics, but most of you are not physicists.", 'start': 1394.881, 'duration': 3.782}, {'end': 1400.504, 'text': "I'm not a physicist.", 'start': 1398.703, 'duration': 1.801}, {'end': 1406.508, 'text': "The reason I'm talking about it is not because of the physics applications, but because it is important for counting.", 'start': 1400.725, 'duration': 5.783}, {'end': 1412.231, 'text': "okay?. But for probability, usually with the naive definition, you're not gonna be able to apply this,", 'start': 1406.508, 'duration': 5.723}, {'end': 1415.312, 'text': "because it's gonna function more like the labeled case, not like this case.", 'start': 1412.231, 'duration': 3.081}, {'end': 1423.157, 'text': "All right, so now it doesn't look like we've done anything except draw a picture, but actually we're basically done deriving the result.", 'start': 1416.233, 'duration': 6.924}], 'summary': 'Physics and counting problems emphasize importance in probability applications.', 'duration': 35.802, 'max_score': 1387.355, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1387355.jpg'}, {'end': 1482.479, 'src': 'embed', 'start': 1453.845, 'weight': 2, 'content': [{'end': 1463.307, 'text': "So I'm gonna draw a vertical line segment to denote the separators between boxes like that.", 'start': 1453.845, 'duration': 9.462}, {'end': 1471.67, 'text': "Okay, so the second box was empty, so therefore I'm drawing two vertical lines without any dots in between.", 'start': 1464.684, 'duration': 6.986}, {'end': 1478.576, 'text': "The third box had two dots and then there's a separator and then there's another dot.", 'start': 1472.07, 'duration': 6.506}, {'end': 1480.097, 'text': 'okay?. So does everyone see what I did?', 'start': 1478.576, 'duration': 1.521}, {'end': 1482.479, 'text': "So it's a very simple encoding.", 'start': 1480.517, 'duration': 1.962}], 'summary': 'Using vertical lines to encode boxes with dots, a simple and clear method.', 'duration': 28.634, 'max_score': 1453.845, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1453845.jpg'}, {'end': 1593.044, 'src': 'heatmap', 'start': 1527.952, 'weight': 0.715, 'content': [{'end': 1534.851, 'text': 'okay?. Now, to specify this, How many ways are there to do this??', 'start': 1527.952, 'duration': 6.899}, {'end': 1538.874, 'text': "Well, you could think of it as like the factorial of all these things, except that that's over counting.", 'start': 1534.931, 'duration': 3.943}, {'end': 1542.796, 'text': "It's similar to the problem like on the strategic practice.", 'start': 1539.314, 'duration': 3.482}, {'end': 1547.239, 'text': "there's a problem like how many ways are there to rearrange the letters of the word pepper right?", 'start': 1542.796, 'duration': 4.443}, {'end': 1550.942, 'text': "So it's not just the factorial of the number of letters in pepper,", 'start': 1547.439, 'duration': 3.503}, {'end': 1555.345, 'text': "because there's multiple Ps and multiple Es and you have to adjust for that over counting.", 'start': 1550.942, 'duration': 4.403}, {'end': 1566.277, 'text': 'Same as that, but an even easier way to think of it is, To specify this, we have n plus k minus 1 positions here.', 'start': 1555.865, 'duration': 10.412}, {'end': 1577.008, 'text': 'And in order to specify our code, all we need to do is specify where are the dots, right? Choose the positions for the dots.', 'start': 1568.179, 'duration': 8.829}, {'end': 1580.071, 'text': 'The remaining positions are the positions for the separators.', 'start': 1577.489, 'duration': 2.582}, {'end': 1583.195, 'text': 'So n plus k minus 1, choose k.', 'start': 1580.872, 'duration': 2.323}, {'end': 1587.16, 'text': "There isn't even anything else I can write on.", 'start': 1584.938, 'duration': 2.222}, {'end': 1588.801, 'text': 'This is self-annotating again.', 'start': 1587.18, 'duration': 1.621}, {'end': 1593.044, 'text': "I have n plus k minus 1 positions here, and I'm going to pick k of them to put the dots.", 'start': 1589.101, 'duration': 3.943}], 'summary': 'To specify positions, choose n plus k minus 1, choose k', 'duration': 65.092, 'max_score': 1527.952, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1527952.jpg'}, {'end': 1587.16, 'src': 'embed', 'start': 1555.865, 'weight': 0, 'content': [{'end': 1566.277, 'text': 'Same as that, but an even easier way to think of it is, To specify this, we have n plus k minus 1 positions here.', 'start': 1555.865, 'duration': 10.412}, {'end': 1577.008, 'text': 'And in order to specify our code, all we need to do is specify where are the dots, right? Choose the positions for the dots.', 'start': 1568.179, 'duration': 8.829}, {'end': 1580.071, 'text': 'The remaining positions are the positions for the separators.', 'start': 1577.489, 'duration': 2.582}, {'end': 1583.195, 'text': 'So n plus k minus 1, choose k.', 'start': 1580.872, 'duration': 2.323}, {'end': 1587.16, 'text': "There isn't even anything else I can write on.", 'start': 1584.938, 'duration': 2.222}], 'summary': 'To specify the code, choose positions for n+k-1, then n+k-1 choose k.', 'duration': 31.295, 'max_score': 1555.865, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1555865.jpg'}, {'end': 1773.684, 'src': 'embed', 'start': 1735.391, 'weight': 5, 'content': [{'end': 1737.313, 'text': 'And this just looks like a simple little counting thing.', 'start': 1735.391, 'duration': 1.922}, {'end': 1739.256, 'text': 'But ideas along these lines in physics.', 'start': 1737.393, 'duration': 1.863}, {'end': 1747.584, 'text': 'they used these to predict a new state of matter called the Bose-Einstein condensate, Which was only they predicted that theoretically,', 'start': 1739.256, 'duration': 8.328}, {'end': 1750.286, 'text': 'it was only empirically observed 70 years later.', 'start': 1747.584, 'duration': 2.702}, {'end': 1753.588, 'text': 'So they predicted it 70 years in advance.', 'start': 1750.366, 'duration': 3.222}, {'end': 1758.85, 'text': "This is not a physics class, but you can look that up if you're curious.", 'start': 1753.608, 'duration': 5.242}, {'end': 1760.851, 'text': 'It has all kinds of bizarre properties.', 'start': 1758.87, 'duration': 1.981}, {'end': 1773.684, 'text': 'The point of this though is that, For coins, thinking of it as labeled, whether you can tell them apart or not, is normally the right way to go.', 'start': 1762.212, 'duration': 11.472}], 'summary': 'Prediction of bose-einstein condensate 70 years in advance using counting methods in physics.', 'duration': 38.293, 'max_score': 1735.391, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1735391.jpg'}, {'end': 1826.287, 'src': 'embed', 'start': 1783.327, 'weight': 4, 'content': [{'end': 1791.549, 'text': "OK So let's talk a little more about counting and what I call story proofs.", 'start': 1783.327, 'duration': 8.222}, {'end': 1799.729, 'text': "So Let's start with a simple example.", 'start': 1794.17, 'duration': 5.559}, {'end': 1811.597, 'text': "So a story proof is still a proof, otherwise I wouldn't call it a proof.", 'start': 1804.312, 'duration': 7.285}, {'end': 1814.019, 'text': 'Someone asked whether that means an example.', 'start': 1811.777, 'duration': 2.242}, {'end': 1826.287, 'text': 'What it means is an application or an interpretation, so proof by interpretation, I would say, rather than proof by algebra or calculus.', 'start': 1815.099, 'duration': 11.188}], 'summary': 'Discussing counting and story proofs, emphasizing interpretation over algebra or calculus.', 'duration': 42.96, 'max_score': 1783.327, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1783327.jpg'}, {'end': 1987.79, 'src': 'embed', 'start': 1955.287, 'weight': 3, 'content': [{'end': 1956.668, 'text': 'I just derived it.', 'start': 1955.287, 'duration': 1.381}, {'end': 1966.275, 'text': 'Again, you can check this by algebra pretty easily.', 'start': 1963.713, 'duration': 2.562}, {'end': 1970.698, 'text': "But that's not going to help you remember it or understand it.", 'start': 1967.416, 'duration': 3.282}, {'end': 1978.763, 'text': "That's just like, you could write out the algebra and it'll just look like a curiosity, it cancels out.", 'start': 1972.798, 'duration': 5.965}, {'end': 1982.486, 'text': "It doesn't give you any intuition for why that's true, okay?", 'start': 1978.823, 'duration': 3.663}, {'end': 1987.79, 'text': "So the story proof for this would be to imagine that we're gonna pick k people.", 'start': 1982.706, 'duration': 5.084}], 'summary': "Algebra alone doesn't aid understanding; story proofs provide intuition.", 'duration': 32.503, 'max_score': 1955.287, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1955287.jpg'}, {'end': 2091.899, 'src': 'heatmap', 'start': 2022.811, 'weight': 0.833, 'content': [{'end': 2028.978, 'text': "So I want to know, how many ways are there to do that? Well, there's two different approaches I could take.", 'start': 2022.811, 'duration': 6.167}, {'end': 2033.06, 'text': "Either I could first select who's in the club.", 'start': 2030.299, 'duration': 2.761}, {'end': 2036.522, 'text': 'There are n people in the population.', 'start': 2033.541, 'duration': 2.981}, {'end': 2037.863, 'text': 'There are k people in my club.', 'start': 2036.542, 'duration': 1.321}, {'end': 2041.304, 'text': "So I could pick who's in the club and choose k.", 'start': 2038.183, 'duration': 3.121}, {'end': 2045.386, 'text': 'And then one of those k must be elected as president, so we multiply by k.', 'start': 2041.304, 'duration': 4.082}, {'end': 2046.267, 'text': 'Multiplication rule.', 'start': 2045.386, 'duration': 0.881}, {'end': 2049.188, 'text': "Choose who's in the club, then choose the president.", 'start': 2047.267, 'duration': 1.921}, {'end': 2050.228, 'text': "That's this.", 'start': 2049.849, 'duration': 0.379}, {'end': 2054.268, 'text': 'But I could also just say, first choose the president.', 'start': 2051.107, 'duration': 3.161}, {'end': 2060.732, 'text': 'And then once I have the president, then I need k minus 1 more people in my club.', 'start': 2055.409, 'duration': 5.323}, {'end': 2063.554, 'text': 'And those could be any of the remaining n minus 1.', 'start': 2060.813, 'duration': 2.741}, {'end': 2064.655, 'text': 'Those, again, the multiplication rule.', 'start': 2063.554, 'duration': 1.101}, {'end': 2065.775, 'text': 'Those are the same thing.', 'start': 2064.995, 'duration': 0.78}, {'end': 2067.356, 'text': "That's a proof.", 'start': 2066.795, 'duration': 0.561}, {'end': 2070.197, 'text': "That's a completely rigorous mathematical proof.", 'start': 2068.137, 'duration': 2.06}, {'end': 2072.438, 'text': 'But it also gives you some interpretation.', 'start': 2070.638, 'duration': 1.8}, {'end': 2076.701, 'text': "So that's the kind of thing that I mean.", 'start': 2074.62, 'duration': 2.081}, {'end': 2081.857, 'text': "That we're counting the same thing in two different ways.", 'start': 2078.735, 'duration': 3.122}, {'end': 2091.899, 'text': "So if both ways are correct, they must agree, right? So that's the idea.", 'start': 2086.518, 'duration': 5.381}], 'summary': 'Two ways to choose club members and president, illustrating counting principle in combinatorics.', 'duration': 69.088, 'max_score': 2022.811, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg2022811.jpg'}, {'end': 2160.875, 'src': 'embed', 'start': 2136.206, 'weight': 6, 'content': [{'end': 2141.89, 'text': 'So suppose we had this sum, and we wanna prove that this sum just collapses just to this one binomial coefficient.', 'start': 2136.206, 'duration': 5.684}, {'end': 2146.974, 'text': 'This is a famous identity in math called Vandermans identity.', 'start': 2142.491, 'duration': 4.483}, {'end': 2155.181, 'text': 'It comes up actually a lot in different areas of math, and especially in probability, but it also comes up outside of probability.', 'start': 2149.356, 'duration': 5.825}, {'end': 2160.875, 'text': "So if you try to derive this one using algebra, it's pretty horrible, right?", 'start': 2156.633, 'duration': 4.242}], 'summary': 'Vandermans identity collapses to one binomial coefficient, used in math and probability.', 'duration': 24.669, 'max_score': 2136.206, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg2136206.jpg'}], 'start': 1387.355, 'title': 'Combinatorics in physics and mathematics', 'summary': "Discusses the significance of combinatorics in physics and mathematics, including a simple encoding method, deriving a formula for specifying positions with n plus k minus 1 positions and n plus k minus 1 choose k possibilities, bose's controversial theory, story proofs in mathematics, and vandermans identity with an intuitive proof, emphasizing the crucial role of combinatorics in various areas of physics and mathematics.", 'chapters': [{'end': 1610.978, 'start': 1387.355, 'title': 'Combinatorics in physics and counting', 'summary': 'Discusses the importance of combinatorics in physics and counting, illustrating a simple encoding method and deriving a formula for specifying positions, with n plus k minus 1 positions and n plus k minus 1 choose k possibilities.', 'duration': 223.623, 'highlights': ['The chapter emphasizes the importance of combinatorics in physics and counting, stating that it is significant for counting problems even though most individuals are not physicists.', 'A simple encoding method using dots and separators is illustrated, providing a visual representation of the concept.', 'The derivation of the formula for specifying positions is explained, with n plus k minus 1 positions and n plus k minus 1 choose k possibilities.']}, {'end': 2097.581, 'start': 1613.608, 'title': "Bose's controversial theory and story proofs in mathematics", 'summary': "Discusses bose's controversial proposal of three equally likely outcomes in physics, leading to the prediction of bose-einstein condensate 70 years before its empirical observation, and emphasizes the use of 'story proofs' for interpreting mathematical concepts and identities.", 'duration': 483.973, 'highlights': ["Bose proposed a model in physics where there were three equally likely outcomes, not four, leading to the prediction of Bose-Einstein condensate 70 years before its empirical observation. Bose's proposal of three equally likely outcomes in physics led to the prediction of Bose-Einstein condensate, 70 years before its empirical observation.", "The use of 'story proofs' in mathematics is emphasized for interpreting mathematical concepts and identities, such as the interpretation of n choose k equals n choose n minus k. The chapter emphasizes the use of 'story proofs' in mathematics for interpreting mathematical concepts and identities, such as the interpretation of n choose k equals n choose n minus k.", "The chapter explains the concept of 'story proofs' as a method of counting the same thing in two different ways, ensuring agreement between both interpretations. The concept of 'story proofs' is explained as a method of counting the same thing in two different ways, ensuring agreement between both interpretations."]}, {'end': 2330.186, 'start': 2104.722, 'title': 'Vandermans identity and its intuitive proof', 'summary': 'Introduces vandermans identity, a famous mathematical identity that can be proven using a story-based approach, making it easier to understand and visualize, which is crucial in various areas of mathematics, including probability.', 'duration': 225.464, 'highlights': ['The chapter introduces Vandermans identity, a famous mathematical identity that can be proven using a story-based approach, making it easier to understand and visualize, which is crucial in various areas of mathematics, including probability.', "The identity m plus n choose k, where it's written as a sum, is shown to collapse to one binomial coefficient, known as Vandermans identity, which is widely used in different areas of mathematics, especially in probability.", 'The proof of Vandermans identity using algebra is deemed difficult and complex, involving factorials, cancellation, and application of the binomial theorem.', 'The chapter provides an intuitive story-based proof of Vandermans identity by visualizing the selection of k people from a group of size m plus n, making it easier to comprehend the mathematical concept and its significance in probability and other areas of mathematics.']}], 'duration': 942.831, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg1387355.jpg', 'highlights': ['The derivation of the formula for specifying positions is explained, with n plus k minus 1 positions and n plus k minus 1 choose k possibilities.', 'The chapter emphasizes the importance of combinatorics in physics and counting, stating that it is significant for counting problems even though most individuals are not physicists.', 'A simple encoding method using dots and separators is illustrated, providing a visual representation of the concept.', "The use of 'story proofs' in mathematics is emphasized for interpreting mathematical concepts and identities, such as the interpretation of n choose k equals n choose n minus k.", "The chapter explains the concept of 'story proofs' as a method of counting the same thing in two different ways, ensuring agreement between both interpretations.", 'Bose proposed a model in physics where there were three equally likely outcomes, not four, leading to the prediction of Bose-Einstein condensate 70 years before its empirical observation.', 'The chapter introduces Vandermans identity, a famous mathematical identity that can be proven using a story-based approach, making it easier to understand and visualize, which is crucial in various areas of mathematics, including probability.', "The identity m plus n choose k, where it's written as a sum, is shown to collapse to one binomial coefficient, known as Vandermans identity, which is widely used in different areas of mathematics, especially in probability.", 'The chapter provides an intuitive story-based proof of Vandermans identity by visualizing the selection of k people from a group of size m plus n, making it easier to comprehend the mathematical concept and its significance in probability and other areas of mathematics.']}, {'end': 2738.62, 'segs': [{'end': 2413.371, 'src': 'embed', 'start': 2375.728, 'weight': 0, 'content': [{'end': 2378.629, 'text': "And we don't wanna have to assume that there are only finitely many possible outcomes.", 'start': 2375.728, 'duration': 2.901}, {'end': 2383.27, 'text': 'So we wanna go beyond that, okay? So this is the non-naive definition.', 'start': 2378.649, 'duration': 4.621}, {'end': 2392.172, 'text': 'For the non-naive definition, we need the notion of a probability space.', 'start': 2385.671, 'duration': 6.501}, {'end': 2398.901, 'text': 'And I already introduced the concept of a sample space.', 'start': 2395.919, 'duration': 2.982}, {'end': 2413.371, 'text': "So a probability space consists of two ingredients, which I'll call S and P.", 'start': 2401.183, 'duration': 12.188}], 'summary': 'Non-naive definition expands beyond finitely many outcomes, requires probability space with s and p.', 'duration': 37.643, 'max_score': 2375.728, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg2375728.jpg'}, {'end': 2531.348, 'src': 'embed', 'start': 2503.098, 'weight': 2, 'content': [{'end': 2517.085, 'text': "So an event is a subset of S, that's the subset symbol input and gives P, Which is a number between 0 and 1,", 'start': 2503.098, 'duration': 13.987}, {'end': 2519.645, 'text': 'because we want probabilities just by convention.', 'start': 2517.085, 'duration': 2.56}, {'end': 2524.206, 'text': 'Standard convention is that we want probabilities to be numbers between 0 and 1.', 'start': 2520.126, 'duration': 4.08}, {'end': 2530.548, 'text': 'So the input is an event, the output is a number between 0 and 1.', 'start': 2524.206, 'duration': 6.342}, {'end': 2531.348, 'text': "So that's the output.", 'start': 2530.548, 'duration': 0.8}], 'summary': 'An event, a subset of s, gives p, a number between 0 and 1 for probabilities.', 'duration': 28.25, 'max_score': 2503.098, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg2503098.jpg'}, {'end': 2589.662, 'src': 'embed', 'start': 2558.565, 'weight': 3, 'content': [{'end': 2574.139, 'text': 'So such that rule number one The probability of the empty set equals 0, and the probability of the full space equals 1.', 'start': 2558.565, 'duration': 15.574}, {'end': 2577.099, 'text': 'And you might say I cheated by including two things in one.', 'start': 2574.139, 'duration': 2.96}, {'end': 2584.361, 'text': 'Actually, you can try to simplify the axioms and you could try to derive this.', 'start': 2577.499, 'duration': 6.862}, {'end': 2587.142, 'text': "But I'm not trying to make this completely minimal.", 'start': 2584.461, 'duration': 2.681}, {'end': 2589.662, 'text': 'I like to write it this way because these are the two extremes.', 'start': 2587.242, 'duration': 2.42}], 'summary': 'Probability of empty set = 0, full space = 1. axioms can be simplified.', 'duration': 31.097, 'max_score': 2558.565, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg2558565.jpg'}], 'start': 2330.346, 'title': 'Probability foundations', 'summary': 'Introduces the non-naive definition of probability, emphasizing the move beyond assuming equally likely outcomes and finite possibilities, and the need for a probability space consisting of a sample space (s) and probability function (p). it also covers the concept of probability as a function that maps events to numbers between 0 and 1, with two axioms to be satisfied: the probability of the empty set equals 0 and the probability of the full space equals 1, forming the basis for all probability theorems and results.', 'chapters': [{'end': 2434.839, 'start': 2330.346, 'title': 'Non-naive definition of probability', 'summary': 'Introduces the non-naive definition of probability, emphasizing the move beyond assuming equally likely outcomes and finite possibilities, and the need for a probability space consisting of a sample space (s) and probability function (p).', 'duration': 104.493, 'highlights': ['The non-naive definition of probability is introduced, emphasizing the move beyond assuming equally likely outcomes and finite possibilities.', 'The concept of a probability space is explained, consisting of a sample space (S) and a probability function (P).', 'S is defined as the sample space, representing the set of all possible outcomes of an experiment, without limitations on equally likely outcomes or finite possibilities.']}, {'end': 2738.62, 'start': 2436.801, 'title': 'Axioms of probability', 'summary': 'Introduces the concept of probability as a function that maps events to numbers between 0 and 1, with two axioms to be satisfied: the probability of the empty set equals 0 and the probability of the full space equals 1. the chapter emphasizes the significance of these simple rules as they form the basis for all probability theorems and results.', 'duration': 301.819, 'highlights': ['Probability is introduced as a function mapping events to numbers between 0 and 1, with the domain of the function being all subsets of S.', 'The axioms of probability include the probability of the empty set equals 0 and the probability of the full space equals 1, serving as the basis for all probability theorems and results.', 'The probability function satisfies the axiom that the probability of the union of non-overlapping events equals the sum of the probabilities of the individual events.']}], 'duration': 408.274, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/FJd_1H3rZGg/pics/FJd_1H3rZGg2330346.jpg', 'highlights': ['The non-naive definition of probability moves beyond assuming equally likely outcomes and finite possibilities.', 'A probability space consists of a sample space (S) and a probability function (P).', 'Probability is a function mapping events to numbers between 0 and 1, with the domain being all subsets of S.', 'The axioms of probability include the probability of the empty set equals 0 and the probability of the full space equals 1.']}], 'highlights': ['The lecture emphasizes the significance of common sense and reasonableness in addressing homework.', 'The chapter explains the process of splitting a group of 10 people into a team of 4 and a team of 6 using combinations.', 'The derivation of the formula for specifying positions is explained, with n plus k minus 1 positions and n plus k minus 1 choose k possibilities.', 'The non-naive definition of probability moves beyond assuming equally likely outcomes and finite possibilities.']}