title

Lecture 1: Probability and Counting | Statistics 110

description

We introduce sample spaces and the naive definition of probability (we'll get to the non-naive definition later). To apply the naive definition, we need to be able to count. So we introduce the multiplication rule, binomial coefficients, and the sampling table (for sampling with/without replacement when order does/doesn't matter).

detail

{'title': 'Lecture 1: Probability and Counting | Statistics 110', 'heatmap': [{'end': 2347.401, 'start': 2260.331, 'weight': 0.718}, {'end': 2454.411, 'start': 2364.955, 'weight': 0.87}], 'summary': 'Covers strategic practice for pattern recognition, clarity in mathematical justifications, math class announcements, and applications of probability in various fields, historical development of probability, mathematical precision, naive probability definition, probability fundamentals, and combinations and binomial coefficients with a focus on poker and full house probability.', 'chapters': [{'end': 227.118, 'segs': [{'end': 29.809, 'src': 'embed', 'start': 0.009, 'weight': 1, 'content': [{'end': 4.011, 'text': 'And the reason I call it strategic practice is kind of that.', 'start': 0.009, 'duration': 4.002}, {'end': 10.256, 'text': "they're gonna be grouped by theme like these are problems that help you practice this topic, these are problems that help you practice that topic,", 'start': 4.011, 'duration': 6.245}, {'end': 10.696, 'text': 'and so on.', 'start': 10.256, 'duration': 0.44}, {'end': 15.94, 'text': "Which is like, as I said, I'm a chess player and my favorite chess tactics books.", 'start': 11.256, 'duration': 4.684}, {'end': 24.005, 'text': "they'll start out with a chapter on pins and then a chapter on forks and a chapter on skewers, or you're just practicing individual chess tactics.", 'start': 15.94, 'duration': 8.065}, {'end': 29.809, 'text': "And then towards the end, you get the chapters that mix everything together, and you don't know whether it's gonna be a fork, or a pin, or a skewer.", 'start': 24.185, 'duration': 5.624}], 'summary': 'Strategic practice involves grouping problems by theme to help practice specific topics, similar to chess tactics books.', 'duration': 29.8, 'max_score': 0.009, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw9.jpg'}, {'end': 61.513, 'src': 'embed', 'start': 35.632, 'weight': 0, 'content': [{'end': 41.575, 'text': "But by doing a lot of problems, you're gonna be improving your pattern recognition skills.", 'start': 35.632, 'duration': 5.943}, {'end': 46.197, 'text': 'So a lot of this course is about pattern recognition, and that just takes practice right?', 'start': 42.055, 'duration': 4.142}, {'end': 50.219, 'text': 'You have to practice as much as you can, so the more practice problems you do, the better.', 'start': 46.537, 'duration': 3.682}, {'end': 56.607, 'text': "I'm not trying to torture anyone with a lot of problems, but as I said last time, this is a difficult course.", 'start': 51.079, 'duration': 5.528}, {'end': 61.513, 'text': 'And the best way to learn it is just by doing a lot of problems.', 'start': 57.828, 'duration': 3.685}], 'summary': 'Practice pattern recognition skills by doing a lot of problems for better learning in a difficult course.', 'duration': 25.881, 'max_score': 35.632, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw35632.jpg'}, {'end': 148.396, 'src': 'embed', 'start': 114.133, 'weight': 3, 'content': [{'end': 122.075, 'text': "Just because this is a mathematical class doesn't mean you shouldn't be using English and explaining things as well as equations.", 'start': 114.133, 'duration': 7.942}, {'end': 130.576, 'text': "So I'd like you to be as clear and detailed as possible in just fully justifying your answers.", 'start': 124.115, 'duration': 6.461}, {'end': 136.478, 'text': 'If the answer is 42, well, the TF knows that the answer is 42.', 'start': 130.776, 'duration': 5.702}, {'end': 143.419, 'text': 'The question is, what reasoning led you to think that? So clarity is a good word.', 'start': 136.478, 'duration': 6.941}, {'end': 148.396, 'text': "And I'd even say honesty.", 'start': 146.675, 'duration': 1.721}], 'summary': 'In a mathematical class, clarity and detailed justifications are emphasized, aiming for honesty in reasoning.', 'duration': 34.263, 'max_score': 114.133, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw114133.jpg'}, {'end': 205.391, 'src': 'embed', 'start': 178.652, 'weight': 5, 'content': [{'end': 184.158, 'text': "But I'd rather say you don't understand it than try to make something up like random gibberish.", 'start': 178.652, 'duration': 5.506}, {'end': 198.207, 'text': "I always think of it as like If I were considering hiring you to build a bridge for me, and I'm thinking of three possibilities either.", 'start': 185.459, 'duration': 12.748}, {'end': 202.35, 'text': "if you just tell me here's the specifications you do this, this and this,", 'start': 198.207, 'duration': 4.143}, {'end': 205.391, 'text': "I'm not gonna have much confidence if you just say that that's the answer right?", 'start': 202.35, 'duration': 3.041}], 'summary': 'Clear communication is vital for confidence in understanding; vague instructions may lead to uncertainty.', 'duration': 26.739, 'max_score': 178.652, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw178652.jpg'}], 'start': 0.009, 'title': 'Strategic practice and clarity in mathematics', 'summary': 'Covers strategic practice for pattern recognition, emphasizing themed problem grouping and the importance of extensive practice. it also highlights the significance of clarity in mathematical justifications, discouraging sloppy arguments and promoting honesty through a compelling analogy of hiring someone to build a bridge.', 'chapters': [{'end': 82.135, 'start': 0.009, 'title': 'Strategic practice for pattern recognition', 'summary': 'Introduces strategic practice for pattern recognition through themed problem grouping, emphasizing the importance of pattern recognition and the need for extensive practice to improve skills.', 'duration': 82.126, 'highlights': ['The chapter emphasizes the importance of pattern recognition and the need for extensive practice to improve skills.', 'Strategic practice involves themed problem grouping to help practice specific topics, gradually integrating them for improved pattern recognition.', 'The course emphasizes a lot of practice problems as the best way to learn and improve pattern recognition skills.']}, {'end': 227.118, 'start': 82.955, 'title': 'Importance of clarity in mathematical justification', 'summary': 'Emphasizes the importance of clarity and detailed reasoning in mathematical justifications, discouraging sloppy arguments and promoting honesty, as exemplified through the analogy of hiring someone to build a bridge.', 'duration': 144.163, 'highlights': ['The significance of clarity and detailed reasoning in mathematical justifications is emphasized, promoting honesty and discouraging sloppy arguments.', 'The importance of using English to explain equations in addition to mathematical expressions is highlighted, aiming for clear and detailed justifications.', 'The analogy of hiring someone to build a bridge is used to illustrate the importance of providing clear reasoning and not resorting to random gibberish in mathematical justifications.']}], 'duration': 227.109, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw9.jpg', 'highlights': ['The chapter emphasizes the importance of pattern recognition and extensive practice.', 'Strategic practice involves themed problem grouping for improved pattern recognition.', 'The course emphasizes a lot of practice problems as the best way to learn and improve skills.', 'The significance of clarity and detailed reasoning in mathematical justifications is emphasized.', 'Using English to explain equations is highlighted for clear and detailed justifications.', 'The analogy of hiring someone to build a bridge illustrates the importance of clear reasoning.']}, {'end': 748.109, 'segs': [{'end': 313.733, 'src': 'embed', 'start': 286.192, 'weight': 1, 'content': [{'end': 289.633, 'text': 'So the math review handout, I posted before.', 'start': 286.192, 'duration': 3.441}, {'end': 291.313, 'text': "But I've made a few updates to it.", 'start': 289.693, 'duration': 1.62}, {'end': 293.954, 'text': 'So you can download a new copy of the math review.', 'start': 291.373, 'duration': 2.581}, {'end': 296.694, 'text': 'And hopefully, most of that material is review.', 'start': 294.054, 'duration': 2.64}, {'end': 299.515, 'text': 'But you should definitely take a look at everything in there.', 'start': 296.774, 'duration': 2.741}, {'end': 301.395, 'text': 'And then today at 2, review.', 'start': 300.195, 'duration': 1.2}, {'end': 313.733, 'text': 'Review sessions are Fridays at 2 in Hall E.', 'start': 307.429, 'duration': 6.304}], 'summary': 'New updates to math review handout, review sessions on fridays at 2 in hall e.', 'duration': 27.541, 'max_score': 286.192, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw286192.jpg'}, {'end': 396.41, 'src': 'embed', 'start': 361.053, 'weight': 2, 'content': [{'end': 365.795, 'text': "Okay, and I also want to mention that, so the homework's due at the beginning of class.", 'start': 361.053, 'duration': 4.742}, {'end': 372.038, 'text': "And there's really no leeway with late homeworks, because first of all, it's such a large course.", 'start': 366.595, 'duration': 5.443}, {'end': 377.04, 'text': "Secondly, I'm gonna post the solutions to the homework very soon after you turn them in.", 'start': 372.078, 'duration': 4.962}, {'end': 380.842, 'text': 'But I do drop the two lowest homework scores.', 'start': 377.72, 'duration': 3.122}, {'end': 385.203, 'text': "So if it's late, it would just become one of your dropped two.", 'start': 381.282, 'duration': 3.921}, {'end': 396.41, 'text': 'Okay, all right, so last time I was kind of quickly mentioning some of the areas where probability is used.', 'start': 387.648, 'duration': 8.762}], 'summary': 'Homework due at start, no leeway for late submissions, 2 lowest scores dropped', 'duration': 35.357, 'max_score': 361.053, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw361053.jpg'}, {'end': 446.611, 'src': 'embed', 'start': 418.834, 'weight': 0, 'content': [{'end': 425.221, 'text': "So last time I just mentioned very briefly in physics, quantum mechanics, it's all probability.", 'start': 418.834, 'duration': 6.387}, {'end': 428.584, 'text': "Genetics, you can't do genetics without probability.", 'start': 425.581, 'duration': 3.003}, {'end': 434.01, 'text': 'And some of the sciences, econometrics and game theory, and so on.', 'start': 428.984, 'duration': 5.026}, {'end': 439.896, 'text': 'But I want to mention a few of the less obvious applications, like history.', 'start': 434.03, 'duration': 5.866}, {'end': 446.611, 'text': 'Well, you might think, what does history and probability have to do with each other?', 'start': 442.668, 'duration': 3.943}], 'summary': 'Probability applies to physics, genetics, econometrics, game theory, and even history.', 'duration': 27.777, 'max_score': 418.834, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw418834.jpg'}, {'end': 724.412, 'src': 'embed', 'start': 698.749, 'weight': 4, 'content': [{'end': 705.095, 'text': 'That the historical roots of the subject are exactly in games of chance, gambling.', 'start': 698.749, 'duration': 6.346}, {'end': 709.279, 'text': 'And so it gives me a chance to talk about a little bit of history.', 'start': 705.916, 'duration': 3.363}, {'end': 718.727, 'text': "It's also some familiar concrete example, like dice and cards and coins and things like that that people gamble with.", 'start': 709.319, 'duration': 9.408}, {'end': 721.81, 'text': 'So I mentioned the math prerequisite before.', 'start': 719.468, 'duration': 2.342}, {'end': 724.412, 'text': "I didn't mention the cards prerequisite.", 'start': 721.83, 'duration': 2.582}], 'summary': 'Historical roots of subject in gambling, with concrete examples like dice, cards, and coins.', 'duration': 25.663, 'max_score': 698.749, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw698749.jpg'}], 'start': 228.538, 'title': 'Math class announcements and applications of probability', 'summary': 'Covers math class announcements including homework style, math review updates, and availability of optional sessions, also explores applications of probability in physics, genetics, history, finance, and gambling, impacting academic and career choices.', 'chapters': [{'end': 385.203, 'start': 228.538, 'title': 'Math class announcements', 'summary': 'Covers announcements for the math class, including the style of homework, updates to the math review handout, and the availability of optional math review sessions on fridays at 2 in hall e, with the reminder that late homework will count towards the two lowest scores.', 'duration': 156.665, 'highlights': ['The homework style requires clarity, honesty, and written explanations, allowing for some leeway with late submissions as the two lowest scores are dropped.', 'Updates have been made to the math review handout, which students are encouraged to thoroughly review.', 'Optional math review sessions are available on Fridays at 2 in Hall E, with a specific math review session happening today, which may be beneficial for those needing to refresh their math skills.', 'The homework is due at the beginning of class with no leeway for late submissions, as the solutions will be posted soon after, and late homework will be counted towards the two lowest scores.']}, {'end': 748.109, 'start': 387.648, 'title': 'Applications of probability in various fields', 'summary': "Explores the diverse applications of probability in fields such as physics, genetics, history, finance, and gambling, including the use of probability in resolving the authorship of the federalist papers and its impact on students' academic and career choices.", 'duration': 360.461, 'highlights': ['Probability is widely used in various fields such as physics, genetics, econometrics, game theory, and history, with notable applications in resolving historical questions like the authorship of the Federalist Papers.', 'A history concentrator, initially with minimal math background, took a statistics course pass-fail and subsequently pursued a statistics-focused academic and career path, illustrating the impact of probability education on academic and career choices.', 'The relevance of probability in finance and the recommendation of a specific statistics course for those interested in finance, emphasizing the practical applications of probability in the financial sector.', 'The historical roots of probability in gambling, with a reference to the use of games of chance as concrete examples, highlighting the historical motivation behind the study of probability.']}], 'duration': 519.571, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw228538.jpg', 'highlights': ["Probability's wide applications in physics, genetics, history, finance, and gambling.", 'Math review updates and availability of optional sessions for students.', 'Homework style allowing leeway with late submissions, dropping two lowest scores.', 'Relevance of probability in finance and its practical applications.', 'Historical roots of probability in gambling and its concrete examples.']}, {'end': 1189.337, 'segs': [{'end': 778.869, 'src': 'embed', 'start': 748.149, 'weight': 2, 'content': [{'end': 753.313, 'text': "And it's also just a source of interesting examples that we can easily explain without getting into a lot of technicalities.", 'start': 748.149, 'duration': 5.164}, {'end': 755.714, 'text': 'And then we can learn things.', 'start': 753.733, 'duration': 1.981}, {'end': 767.445, 'text': "I'm just going to mention Fermat and Pascal in the mid-1650s.", 'start': 759.582, 'duration': 7.863}, {'end': 775.668, 'text': 'And there are other examples as well, but this is arguably the most important kind of historical route of probability.', 'start': 769.486, 'duration': 6.182}, {'end': 778.869, 'text': "So Fermat, you've probably heard of Fermat's Last Theorem.", 'start': 776.048, 'duration': 2.821}], 'summary': 'Exploring historical probability through examples like fermat and pascal in the mid-1650s.', 'duration': 30.72, 'max_score': 748.149, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw748149.jpg'}, {'end': 834.341, 'src': 'embed', 'start': 802.928, 'weight': 5, 'content': [{'end': 806.389, 'text': 'So they were writing very long letters back and forth to each other.', 'start': 802.928, 'duration': 3.461}, {'end': 808.249, 'text': 'And most of the letters have survived.', 'start': 806.709, 'duration': 1.54}, {'end': 812.85, 'text': 'You can actually find them online if you just look up, you know, Fermat, Pascal correspondence.', 'start': 808.269, 'duration': 4.581}, {'end': 815.931, 'text': "It's pretty interesting to read their letters back and forth.", 'start': 813.551, 'duration': 2.38}, {'end': 820.473, 'text': 'And they were just writing letters back and forth to each other, analyzing different gambling games.', 'start': 816.191, 'duration': 4.282}, {'end': 824.254, 'text': "And it's like if you have this gambling game, what's the probability that this will happen?", 'start': 820.493, 'duration': 3.761}, {'end': 825.554, 'text': "What's the probability that that will happen?", 'start': 824.274, 'duration': 1.28}, {'end': 827.535, 'text': 'And that was all completely new at the time.', 'start': 825.914, 'duration': 1.621}, {'end': 834.341, 'text': 'No one had mathematically derived these rules and how to work with probability.', 'start': 828.296, 'duration': 6.045}], 'summary': 'Fermat and pascal exchanged long letters analyzing gambling games, pioneering probability theory.', 'duration': 31.413, 'max_score': 802.928, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw802928.jpg'}, {'end': 911.926, 'src': 'embed', 'start': 874.816, 'weight': 0, 'content': [{'end': 876.637, 'text': 'Everyone has a lot of uncertainties.', 'start': 874.816, 'duration': 1.821}, {'end': 885.182, 'text': 'And probability and statistics are how we quantify and update our beliefs and deal with uncertainty.', 'start': 876.757, 'duration': 8.425}, {'end': 888.385, 'text': "OK, so that's what this course is going to be about.", 'start': 886.103, 'duration': 2.282}, {'end': 891.847, 'text': "It's going to be about quantifying uncertainty.", 'start': 888.985, 'duration': 2.862}, {'end': 902.04, 'text': 'All right, so now we can get to the naive definition of probability, which was the origins of the subject.', 'start': 893.035, 'duration': 9.005}, {'end': 911.926, 'text': "So let me tell you what a sample space is first, then I'll give you the naive definition.", 'start': 905.322, 'duration': 6.604}], 'summary': 'Quantifying uncertainty through probability and statistics. course focuses on naive definition of probability and sample space.', 'duration': 37.11, 'max_score': 874.816, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw874816.jpg'}, {'end': 1045.394, 'src': 'embed', 'start': 937.541, 'weight': 1, 'content': [{'end': 940.964, 'text': 'Do anything as long as there are certain possible outcomes.', 'start': 937.541, 'duration': 3.423}, {'end': 945.107, 'text': "So, before the experiment, you don't know what's going to happen, because there are different possible outcomes,", 'start': 941.304, 'duration': 3.803}, {'end': 946.488, 'text': "and you don't know which one's going to happen.", 'start': 945.107, 'duration': 1.381}, {'end': 949.27, 'text': 'You do the experiment and then something happens.', 'start': 947.108, 'duration': 2.162}, {'end': 956.236, 'text': 'Okay, so what I just said was very, very general, right? I mean, that could describe any number of situations.', 'start': 949.89, 'duration': 6.346}, {'end': 961.761, 'text': 'So sample space is the set of all possible outcomes of an experiment.', 'start': 956.757, 'duration': 5.004}, {'end': 967.407, 'text': 'And we can interpret experiment however we want.', 'start': 964.684, 'duration': 2.723}, {'end': 968.868, 'text': 'So this is a very general concept.', 'start': 967.447, 'duration': 1.421}, {'end': 977.217, 'text': 'all possible outcomes of an experiment.', 'start': 974.696, 'duration': 2.521}, {'end': 982.379, 'text': "And we might say that it's a random experiment, but I'm not gonna use the word random right now cuz we haven't defined it.", 'start': 977.457, 'duration': 4.922}, {'end': 984.86, 'text': "I'm just interpreting this very, very generally.", 'start': 982.679, 'duration': 2.181}, {'end': 991.162, 'text': 'Okay, and then we need one more concept, which is that of an event.', 'start': 986.38, 'duration': 4.782}, {'end': 995.224, 'text': "And we'll come back to this, but the earlier you start thinking about events, the better.", 'start': 991.242, 'duration': 3.982}, {'end': 1000.966, 'text': 'An event is a subset of the sample space.', 'start': 998.225, 'duration': 2.741}, {'end': 1017.154, 'text': "And by the way, there's also a one-page handout that's on the course web page called Probability in Sets or something like that.", 'start': 1009.189, 'duration': 7.965}, {'end': 1027.381, 'text': 'One of the big breakthroughs in probability that made it possible to actually treat this as a mathematical subject instead of just something more like astrology was.', 'start': 1017.754, 'duration': 9.627}, {'end': 1031.151, 'text': 'that was the idea of using sets.', 'start': 1027.381, 'duration': 3.77}, {'end': 1036.452, 'text': 'So most of you have seen unions and intersections and things like that.', 'start': 1032.57, 'duration': 3.882}, {'end': 1045.394, 'text': "But if you don't know much basic facts about set theory like that, I put a short introduction into the review handout.", 'start': 1036.771, 'duration': 8.623}], 'summary': 'A sample space represents all possible outcomes of an experiment, and an event is a subset of the sample space.', 'duration': 107.853, 'max_score': 937.541, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw937541.jpg'}, {'end': 1165.229, 'src': 'embed', 'start': 1140.26, 'weight': 7, 'content': [{'end': 1149.063, 'text': "And so one thing that makes this subject difficult is that we're gonna do a lot of things that are deeply, deeply counterintuitive to almost everyone.", 'start': 1140.26, 'duration': 8.803}, {'end': 1152.684, 'text': 'And I think that makes this a fun subject.', 'start': 1149.903, 'duration': 2.781}, {'end': 1154.465, 'text': "It's a lot of fun teaching that.", 'start': 1153.185, 'duration': 1.28}, {'end': 1158.386, 'text': "There are a lot of paradoxes we'll talk about, a lot of very surprising results.", 'start': 1154.485, 'duration': 3.901}, {'end': 1163.088, 'text': 'So to me, that makes this more fun than calculus.', 'start': 1158.947, 'duration': 4.141}, {'end': 1165.229, 'text': 'When you take a calculus class.', 'start': 1163.228, 'duration': 2.001}], 'summary': 'Teaching this subject involves counterintuitive concepts and paradoxes, making it more enjoyable than calculus.', 'duration': 24.969, 'max_score': 1140.26, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1140260.jpg'}], 'start': 748.149, 'title': "Probability's historical development and understanding sample space", 'summary': "Highlights the historical development of probability, focusing on fermat and pascal's analysis of gambling games, and also discusses the concept of sample space, emphasizing the importance of understanding events, set theory concepts, and the historical significance of probability.", 'chapters': [{'end': 902.04, 'start': 748.149, 'title': 'Historical development of probability', 'summary': 'Highlights the historical development of probability, focusing on the correspondence between fermat and pascal in mid-1650s, their analysis of gambling games, and the significance of probability and statistics in quantifying uncertainty.', 'duration': 153.891, 'highlights': ['The significance of probability and statistics in quantifying uncertainty', "Fermat and Pascal's correspondence and analysis of gambling games", 'The naive definition of probability as the origins of the subject']}, {'end': 1189.337, 'start': 905.322, 'title': 'Understanding sample space and probability', 'summary': 'Discusses the concept of sample space, defining it as the set of all possible outcomes of an experiment, and emphasizes the importance of understanding events and applying set theory concepts like unions, intersections, and complements in probability calculations, highlighting the counterintuitive nature of the subject and its historical significance.', 'duration': 284.015, 'highlights': ["The concept of sample space is introduced as the set of all possible outcomes of an experiment, emphasizing the broad interpretation of 'experiment' and the existence of different possible outcomes before conducting the experiment.", 'The importance of understanding events as subsets of the sample space is highlighted, emphasizing the early consideration of events and their relationship to probability calculations.', 'The significant role of set theory in probability, particularly the use of unions, intersections, and complements, is emphasized as a breakthrough that enabled the mathematical treatment of probability and distinguished it from intuitive reasoning and heuristics.', 'The historical significance of probability is discussed, noting how 300 years ago, calculations that can now be performed in a statistics class required consultation with distinguished figures like Isaac Newton, highlighting the progress and accessibility of probability calculations over time.', "The subject's counterintuitive nature is highlighted, as it is mentioned that many concepts and results in statistics are deeply counterintuitive and lead to surprising and paradoxical outcomes, setting it apart from subjects like calculus."]}], 'duration': 441.188, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw748149.jpg', 'highlights': ['The significance of probability and statistics in quantifying uncertainty', 'The concept of sample space as the set of all possible outcomes', 'The historical significance of probability and its progress over time', 'The significant role of set theory in enabling mathematical treatment of probability', 'The importance of understanding events as subsets of the sample space', "Fermat and Pascal's correspondence and analysis of gambling games", 'The naive definition of probability as the origins of the subject', "The subject's counterintuitive nature and paradoxical outcomes", "The broad interpretation of 'experiment' and the existence of different possible outcomes"]}, {'end': 1610.861, 'segs': [{'end': 1256.779, 'src': 'embed', 'start': 1189.938, 'weight': 0, 'content': [{'end': 1192.12, 'text': 'Which just means you have to work hard, you have to think hard.', 'start': 1189.938, 'duration': 2.182}, {'end': 1196.343, 'text': 'And it will become more intuitive the more you think about it, the more problems you solve.', 'start': 1192.32, 'duration': 4.023}, {'end': 1201.047, 'text': 'But at first, a lot of this might seem counterintuitive, even to Isaac Newton.', 'start': 1196.944, 'duration': 4.103}, {'end': 1210.564, 'text': "OK So that's why we need to be more mathematically precise about it, because our intuitions can easily be completely wrong in probability.", 'start': 1202.208, 'duration': 8.356}, {'end': 1213.228, 'text': "So that's why we need to make it more mathematical.", 'start': 1210.845, 'duration': 2.383}, {'end': 1220.257, 'text': 'And then, as I said, the breakthrough, mathematically speaking, was to start thinking of events as subsets.', 'start': 1213.728, 'duration': 6.529}, {'end': 1224.482, 'text': "Okay, so I'm gonna draw a lot of Venn diagrams in this class.", 'start': 1221.178, 'duration': 3.304}, {'end': 1229.508, 'text': "Usually I'll call the sample space capital S, and it's just a set.", 'start': 1225.803, 'duration': 3.705}, {'end': 1234.133, 'text': 'The elements of the set are possible outcomes of the experiment?', 'start': 1230.449, 'duration': 3.684}, {'end': 1244.943, 'text': 'okay?. So if our experiment is to roll two dice, six-sided dice, then there are 36 possible outcomes.', 'start': 1234.133, 'duration': 10.81}, {'end': 1248.348, 'text': "We'll actually get to where the 36 comes from in a bit.", 'start': 1245.444, 'duration': 2.904}, {'end': 1252.633, 'text': "There's 36 possible outcomes, and then this set would consist of all those outcomes.", 'start': 1248.368, 'duration': 4.265}, {'end': 1256.779, 'text': "And then an event, let's call it A, is just some subset.", 'start': 1253.074, 'duration': 3.705}], 'summary': 'Probability concepts are counterintuitive, requiring mathematical precision. events are treated as subsets, with 36 possible outcomes when rolling two dice.', 'duration': 66.841, 'max_score': 1189.938, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1189938.jpg'}, {'end': 1501.043, 'src': 'embed', 'start': 1465.64, 'weight': 3, 'content': [{'end': 1467.941, 'text': "There are all kinds of different possibilities that we'll consider.", 'start': 1465.64, 'duration': 2.301}, {'end': 1471.402, 'text': 'But the most naive way to write this down would be, okay, we have these four cases.', 'start': 1468.441, 'duration': 2.961}, {'end': 1476.443, 'text': 'If we treat them all as equally likely, then we want the probability of some event.', 'start': 1471.742, 'duration': 4.701}, {'end': 1480.024, 'text': 'We just count how many of those happen divided by a number of things.', 'start': 1476.783, 'duration': 3.241}, {'end': 1481.905, 'text': "That's the naive definition.", 'start': 1480.805, 'duration': 1.1}, {'end': 1485.026, 'text': 'So it has a huge assumption.', 'start': 1483.225, 'duration': 1.801}, {'end': 1491.168, 'text': 'It assumes that all outcomes are equally likely.', 'start': 1488.247, 'duration': 2.921}, {'end': 1501.043, 'text': 'And it also assumes that there are finitely many outcomes.', 'start': 1498.002, 'duration': 3.041}], 'summary': 'Discussing naive definition of probability with assumptions of equal likelihood and finitely many outcomes.', 'duration': 35.403, 'max_score': 1465.64, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1465640.jpg'}, {'end': 1596.229, 'src': 'embed', 'start': 1568.727, 'weight': 5, 'content': [{'end': 1573.948, 'text': "if I wanna know what's the probability that there is life on Neptune?", 'start': 1568.727, 'duration': 5.221}, {'end': 1576.729, 'text': "okay? Well, I've never been there.", 'start': 1573.948, 'duration': 2.781}, {'end': 1579.369, 'text': "I don't know if there's life on Neptune or not.", 'start': 1576.949, 'duration': 2.42}, {'end': 1581.51, 'text': "So either there is or there isn't, that's two.", 'start': 1579.509, 'duration': 2.001}, {'end': 1586.989, 'text': "One of them has life, the other one doesn't, so it'll be one half.", 'start': 1583.816, 'duration': 3.173}, {'end': 1591.245, 'text': "Okay, so most people would agree that's a ridiculous argument.", 'start': 1587.742, 'duration': 3.503}, {'end': 1596.229, 'text': 'Despite that, you can find many examples in the media, in the news.', 'start': 1591.765, 'duration': 4.464}], 'summary': 'Probability of life on neptune: 1/2; media misrepresents ridiculous arguments.', 'duration': 27.502, 'max_score': 1568.727, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1568727.jpg'}], 'start': 1189.938, 'title': 'Mathematical precision in probability and naive probability definition', 'summary': 'Emphasizes the need for mathematical precision in probability and discusses the naive probability definition, illustrated through examples such as rolling two six-sided dice with 36 possible outcomes and coin tosses.', 'chapters': [{'end': 1256.779, 'start': 1189.938, 'title': 'Mathematical precision in probability', 'summary': 'Discusses the need for mathematical precision in probability, emphasizing the importance of working hard and thinking hard to overcome counterintuitive aspects, with a breakthrough in thinking of events as subsets, illustrated through the example of rolling two six-sided dice resulting in 36 possible outcomes.', 'duration': 66.841, 'highlights': ['The breakthrough in probability was to start thinking of events as subsets, emphasizing the need for mathematical precision.', 'The importance of working hard and thinking hard to overcome counterintuitive aspects in probability.', 'Illustration of 36 possible outcomes when rolling two six-sided dice, emphasizing the concept of events as subsets.']}, {'end': 1610.861, 'start': 1257.935, 'title': 'Naive probability definition', 'summary': 'Discusses the naive definition of probability, which assumes all outcomes are equally likely and requires a finite sample space, using an example of coin tosses to illustrate the concept.', 'duration': 352.926, 'highlights': ['The naive definition of probability states that it assumes all outcomes are equally likely and requires a finite sample space, which is a very strong assumption in many cases.', 'An example of a fair coin toss is used to illustrate the naive definition, where the probability is calculated by counting the favorable outcomes and dividing by the total possible outcomes.', 'The discussion highlights the limitations of the naive definition, pointing out that it may lead to absurd conclusions when applied to complex scenarios, such as determining the probability of life on Neptune.']}], 'duration': 420.923, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1189938.jpg', 'highlights': ['The breakthrough in probability was to start thinking of events as subsets, emphasizing the need for mathematical precision.', 'Illustration of 36 possible outcomes when rolling two six-sided dice, emphasizing the concept of events as subsets.', 'The importance of working hard and thinking hard to overcome counterintuitive aspects in probability.', 'The naive definition of probability assumes all outcomes are equally likely and requires a finite sample space, which is a very strong assumption in many cases.', 'An example of a fair coin toss is used to illustrate the naive definition, where the probability is calculated by counting the favorable outcomes and dividing by the total possible outcomes.', 'The discussion highlights the limitations of the naive definition, pointing out that it may lead to absurd conclusions when applied to complex scenarios, such as determining the probability of life on Neptune.']}, {'end': 2134.958, 'segs': [{'end': 1636.824, 'src': 'embed', 'start': 1611.061, 'weight': 2, 'content': [{'end': 1617.087, 'text': "There's no justification for using the naive definition in that case, right? And the situation gets even worse.", 'start': 1611.061, 'duration': 6.026}, {'end': 1620.51, 'text': 'I just said, well, according to this, the probability of life on Neptune is one half.', 'start': 1617.127, 'duration': 3.383}, {'end': 1631.381, 'text': "And what if I asked you instead, what's the probability that there's intelligent life on Neptune? Well, again, either there is or there isn't.", 'start': 1621.051, 'duration': 10.33}, {'end': 1633.543, 'text': 'So that would also be one half.', 'start': 1632.202, 'duration': 1.341}, {'end': 1636.824, 'text': 'But there seems something severely wrong with.', 'start': 1634.203, 'duration': 2.621}], 'summary': 'Probability of life on neptune is one half, but seems severely wrong.', 'duration': 25.763, 'max_score': 1611.061, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1611061.jpg'}, {'end': 1783.899, 'src': 'embed', 'start': 1751.696, 'weight': 0, 'content': [{'end': 1761.374, 'text': "And it says It's a pretty simple principle, but it underlies most of what we'll need for counting,", 'start': 1751.696, 'duration': 9.678}, {'end': 1766.616, 'text': "except for one other counting method that we'll get to next time.", 'start': 1761.374, 'duration': 5.242}, {'end': 1781.537, 'text': "Multiplication rule says that if we have an experiment Again I'm gonna say this kind of abstractly, because this is a general principle,", 'start': 1768.297, 'duration': 13.24}, {'end': 1783.899, 'text': "and then we'll see examples of how to use this.", 'start': 1781.537, 'duration': 2.362}], 'summary': 'Multiplication rule underlies most counting methods.', 'duration': 32.203, 'max_score': 1751.696, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1751696.jpg'}, {'end': 2091.404, 'src': 'embed', 'start': 2063.199, 'weight': 1, 'content': [{'end': 2069.621, 'text': 'Six is still a pretty small number, but if you imagine that there are many branches and each time it keeps branching different ways.', 'start': 2063.199, 'duration': 6.422}, {'end': 2074.165, 'text': "And if you multiply 2 times 2 times 2 many times, it's gonna grow exponentially fast.", 'start': 2069.661, 'duration': 4.504}, {'end': 2076.285, 'text': '2 to the 10th power is 1, 024.', 'start': 2074.725, 'duration': 1.56}, {'end': 2082.149, 'text': "So if we had 10 choices, and each choice we can only choose between two things, there's still over 1, 000 possibilities.", 'start': 2076.286, 'duration': 5.863}, {'end': 2084.63, 'text': 'So these grow very, very, very fast.', 'start': 2082.57, 'duration': 2.06}, {'end': 2091.404, 'text': "And that's why it's hopeless to try to just list them out except for the very simplest problems.", 'start': 2085.478, 'duration': 5.926}], 'summary': 'The number of possibilities grows exponentially fast, with 2 to the 10th power yielding over 1,024 possibilities, making it impossible to list them all.', 'duration': 28.205, 'max_score': 2063.199, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw2063199.jpg'}], 'start': 1611.061, 'title': 'Probability fundamentals', 'summary': 'Discusses the limitations of naive probability definitions and emphasizes the need for a nuanced approach in assessing probabilities in unequal likelihood situations, alongside fundamental counting principles and the application of the multiplication rule in determining possible outcomes for combined experiments, illustrated through examples and explanations.', 'chapters': [{'end': 1692.801, 'start': 1611.061, 'title': 'Probability and naive definitions', 'summary': 'Discusses the limitations of the naive definition of probability, highlighting the need for a more nuanced approach when assessing probabilities in situations with unequal likelihoods, as exemplified by the probability of life on neptune.', 'duration': 81.74, 'highlights': ['The definition of probability based on equally likely outcomes is inadequate for assessing probabilities in scenarios with unequal likelihoods, as seen in the case of the probability of intelligent life on Neptune.', 'The concept of probability evolved from its origins in gambling, emphasizing its historical significance and relevance to a wide range of problems where equal likelihood can be assumed.', 'A clear justification is essential when assuming equally likely outcomes, particularly in cases with a non-finite number of outcomes, as highlighted by the limitations of the naive definition in assessing probabilities.', 'The need for a more sophisticated approach to probability becomes evident when considering scenarios where strict inequalities in likelihood exist, such as the distinction between the probability of any kind of life and the probability of intelligent life on Neptune.']}, {'end': 2134.958, 'start': 1693.262, 'title': 'Counting principles in probability', 'summary': 'Discusses the fundamental principles of counting in probability, emphasizing the multiplication rule and its application in determining overall possible outcomes for combined experiments, presented through an ice cream example and exponential growth explanation.', 'duration': 441.696, 'highlights': ['The multiplication rule states that overall there are n1 times, n2 times, blah, blah, blah times nr overall possible outcomes for combined experiments, illustrated through a simple ice cream example with 6 possibilities.', 'Exponential growth is demonstrated through the example of 2 to the 10th power resulting in over 1,024 possibilities, emphasizing the rapid expansion of possibilities in sequential choices.', 'The futility of listing out possibilities is emphasized due to the exponential growth of choices, except for the very simplest problems, stressing the complexity in enumerating outcomes.']}], 'duration': 523.897, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw1611061.jpg', 'highlights': ['The multiplication rule states n1 times, n2 times, blah, blah, blah times nr overall possible outcomes for combined experiments, illustrated through a simple ice cream example with 6 possibilities.', 'Exponential growth is demonstrated through the example of 2 to the 10th power resulting in over 1,024 possibilities, emphasizing the rapid expansion of possibilities in sequential choices.', 'The definition of probability based on equally likely outcomes is inadequate for assessing probabilities in scenarios with unequal likelihoods, as seen in the case of the probability of intelligent life on Neptune.', 'The need for a more sophisticated approach to probability becomes evident when considering scenarios where strict inequalities in likelihood exist, such as the distinction between the probability of any kind of life and the probability of intelligent life on Neptune.']}, {'end': 2788.585, 'segs': [{'end': 2197.59, 'src': 'embed', 'start': 2135.718, 'weight': 0, 'content': [{'end': 2138.06, 'text': "But I'm gonna use the naive definition of probability.", 'start': 2135.718, 'duration': 2.342}, {'end': 2145.626, 'text': "I wanna know what's the number of possible hands? Well, that's 52 choose 5.", 'start': 2139.641, 'duration': 5.985}, {'end': 2148.548, 'text': 'I think most of you have seen this.', 'start': 2145.626, 'duration': 2.922}, {'end': 2153.212, 'text': 'Sometimes people write this as like 52C5, combinations and things.', 'start': 2149.269, 'duration': 3.943}, {'end': 2155.414, 'text': 'This is a preferable notation.', 'start': 2153.552, 'duration': 1.862}, {'end': 2159.557, 'text': "I'll remind you what it is in case you haven't seen it, but hopefully most of you have seen that before.", 'start': 2156.014, 'duration': 3.543}, {'end': 2163.555, 'text': "So we'll be seeing a lot of these in this course.", 'start': 2160.812, 'duration': 2.743}, {'end': 2165.357, 'text': 'This is called a binomial coefficient.', 'start': 2163.575, 'duration': 1.782}, {'end': 2171.763, 'text': "And it's pronounced n choose k and written like that.", 'start': 2168.76, 'duration': 3.003}, {'end': 2179.871, 'text': "And it's defined as n factorial over n minus k factorial k factorial.", 'start': 2172.704, 'duration': 7.167}, {'end': 2188.947, 'text': "I'm assuming you've seen factorials before, but if not, definitely should make sure you know what factorials are.", 'start': 2181.525, 'duration': 7.422}, {'end': 2197.59, 'text': "We'll also define this as 0 if k is greater than n.", 'start': 2192.768, 'duration': 4.822}], 'summary': 'The number of possible hands is 52 choose 5, denoted as 52c5.', 'duration': 61.872, 'max_score': 2135.718, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw2135718.jpg'}, {'end': 2361.828, 'src': 'heatmap', 'start': 2260.331, 'weight': 2, 'content': [{'end': 2267.074, 'text': "Let's choose the first person, okay? So we have n people and we wanna select k of them, okay? Pick the first person.", 'start': 2260.331, 'duration': 6.743}, {'end': 2269.734, 'text': 'There are n choices, right? Cuz you can pick anyone.', 'start': 2267.694, 'duration': 2.04}, {'end': 2274.596, 'text': 'Now, and then the next person could be anyone except the one you already chose.', 'start': 2270.175, 'duration': 4.421}, {'end': 2275.856, 'text': "So it's gonna be n-1.", 'start': 2274.616, 'duration': 1.24}, {'end': 2277.957, 'text': "And then the next one, there's n-2.", 'start': 2276.677, 'duration': 1.28}, {'end': 2280.098, 'text': 'And it goes all the way like that.', 'start': 2278.857, 'duration': 1.241}, {'end': 2287.272, 'text': 'Until it goes down to n- k plus 1, because if k is 1, I wanna stop at n.', 'start': 2281.178, 'duration': 6.094}, {'end': 2291.339, 'text': 'If k is 2, I wanna stop at n- 1.', 'start': 2287.272, 'duration': 4.067}, {'end': 2292.4, 'text': 'So that would be the answer.', 'start': 2291.339, 'duration': 1.061}, {'end': 2295.561, 'text': 'if we were picking people in a specific order, okay?', 'start': 2292.4, 'duration': 3.161}, {'end': 2301.304, 'text': 'But these k people I just selected, I could have chosen them in any order, right?', 'start': 2295.941, 'duration': 5.363}, {'end': 2304.405, 'text': 'So I have to divide this by k factorial cuz.', 'start': 2301.624, 'duration': 2.781}, {'end': 2306.226, 'text': "I'm over, counted by that factor.", 'start': 2304.405, 'duration': 1.821}, {'end': 2312.849, 'text': "And that's actually the same thing as n factorial over n- k factorial, k factorial.", 'start': 2306.886, 'duration': 5.963}, {'end': 2318.369, 'text': 'Because if you write out these factorials, all the stuff is going to cancel.', 'start': 2313.925, 'duration': 4.444}, {'end': 2320.03, 'text': "And this is what's left.", 'start': 2318.489, 'duration': 1.541}, {'end': 2324.314, 'text': 'If you imagine, this is n times n minus 1, n minus 2, all the way down to 1.', 'start': 2320.09, 'duration': 4.224}, {'end': 2327.036, 'text': 'This one, the same thing starting at n minus k.', 'start': 2324.314, 'duration': 2.722}, {'end': 2327.757, 'text': 'Cancel stuff.', 'start': 2327.036, 'duration': 0.721}, {'end': 2328.477, 'text': "This is what's left.", 'start': 2327.797, 'duration': 0.68}, {'end': 2331.72, 'text': "So that's where this thing comes from.", 'start': 2329.118, 'duration': 2.602}, {'end': 2333.742, 'text': 'Now coming back to this full house problem.', 'start': 2332.02, 'duration': 1.722}, {'end': 2341.418, 'text': 'A full house is defined as having three cards of one rank and two of another.', 'start': 2335.655, 'duration': 5.763}, {'end': 2347.401, 'text': "For example, three sevens and two tens, okay? That's called a full house.", 'start': 2341.778, 'duration': 5.623}, {'end': 2354.765, 'text': 'So if we use the naive definition, which is justified if we assume that the cards are completely shuffled.', 'start': 2349.182, 'duration': 5.583}, {'end': 2361.828, 'text': "The denominator is 52 choose 5 because I'm just choosing five cards out of 52 with all possibilities equally likely.", 'start': 2355.405, 'duration': 6.423}], 'summary': 'Selecting k people from n in specific order with combinations', 'duration': 101.497, 'max_score': 2260.331, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw2260331.jpg'}, {'end': 2454.411, 'src': 'heatmap', 'start': 2364.955, 'weight': 0.87, 'content': [{'end': 2373.96, 'text': "For the numerator, so I'll just say for full house, then I'm just gonna write as an example, three sevens and two tens.", 'start': 2364.955, 'duration': 9.005}, {'end': 2380.124, 'text': 'It really helps to just have some concrete example in mind, some numbers to think about.', 'start': 2374.801, 'duration': 5.323}, {'end': 2381.665, 'text': "okay?. So what's the probability??", 'start': 2380.124, 'duration': 1.541}, {'end': 2384.086, 'text': 'Well, first of all, I need to choose.', 'start': 2382.125, 'duration': 1.961}, {'end': 2391.46, 'text': 'What do I have three of? I wrote down sevens here, but that could have been anything.', 'start': 2385.976, 'duration': 5.484}, {'end': 2395.262, 'text': 'So there are 13 possibilities, or you could say 13 choose one.', 'start': 2392.12, 'duration': 3.142}, {'end': 2400.287, 'text': "Now I'm multiplying because I'm using the multiplication rule, okay?", 'start': 2396.843, 'duration': 3.444}, {'end': 2405.752, 'text': "So in my mind I'm imagining I'm not gonna draw the whole tree because I'd have to draw 13 branches.", 'start': 2400.347, 'duration': 5.405}, {'end': 2411.638, 'text': "But I'm imagining that at the first branching, I'm choosing any rank, ace, two, three, four, whatever.", 'start': 2405.792, 'duration': 5.846}, {'end': 2413.881, 'text': "Now in my mind, I'm imagining I chose seven.", 'start': 2411.999, 'duration': 1.882}, {'end': 2418.125, 'text': "So I'm focusing on the seven branch, and then it's branching further.", 'start': 2414.201, 'duration': 3.924}, {'end': 2419.006, 'text': 'So I have 13 choices.', 'start': 2418.145, 'duration': 0.861}, {'end': 2422.335, 'text': 'and I have sevens, and I need three sevens.', 'start': 2420.174, 'duration': 2.161}, {'end': 2427.539, 'text': 'Well, there are four sevens in a deck of cards, and I need to choose three out of the four.', 'start': 2422.956, 'duration': 4.583}, {'end': 2429.84, 'text': "So I'm going to multiply by four choose three.", 'start': 2427.579, 'duration': 2.261}, {'end': 2432.502, 'text': 'Then I need to choose what the other one is.', 'start': 2430.921, 'duration': 1.581}, {'end': 2435.103, 'text': 'So I wrote tens here, but it could have been anything.', 'start': 2432.582, 'duration': 2.521}, {'end': 2439.166, 'text': "There's 12 possibilities, because it could be anything except sevens in that case.", 'start': 2435.303, 'duration': 3.863}, {'end': 2440.987, 'text': 'And then we need two of those.', 'start': 2439.686, 'duration': 1.301}, {'end': 2444.129, 'text': "So four choose two, and that's it.", 'start': 2441.287, 'duration': 2.842}, {'end': 2452.37, 'text': 'Now, there are other ways to write the answer, OK? But I recommend thinking about it this way, thinking in terms of the tree.', 'start': 2445.626, 'duration': 6.744}, {'end': 2454.411, 'text': "It's a more structured way to do it.", 'start': 2452.89, 'duration': 1.521}], 'summary': 'Calculating the probability of getting a full house with three sevens and two tens in a deck of cards using combinatorics and the multiplication rule.', 'duration': 89.456, 'max_score': 2364.955, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw2364955.jpg'}, {'end': 2600.861, 'src': 'embed', 'start': 2564.895, 'weight': 4, 'content': [{'end': 2568.295, 'text': 'Without replacement would mean then the next person we pick has to be someone different.', 'start': 2564.895, 'duration': 3.4}, {'end': 2570.316, 'text': "So there's two different applications.", 'start': 2568.535, 'duration': 1.781}, {'end': 2574.957, 'text': 'Sometimes the replacement is relevant and sometimes not replacement.', 'start': 2570.376, 'duration': 4.581}, {'end': 2576.817, 'text': 'So sampling with or without replacement.', 'start': 2574.977, 'duration': 1.84}, {'end': 2586.874, 'text': "Okay, and then the other possibility, do we care about order or not? So I'll say order matters or order doesn't matter.", 'start': 2578.729, 'duration': 8.145}, {'end': 2600.861, 'text': "And I wanna fill in this table, okay? Does everyone understand the setup of the problem? We have n objects or n people, we're gonna pick k of them.", 'start': 2592.397, 'duration': 8.464}], 'summary': 'Discussing sampling with or without replacement and order matters or not in picking k from n objects.', 'duration': 35.966, 'max_score': 2564.895, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw2564895.jpg'}, {'end': 2740.464, 'src': 'embed', 'start': 2711.614, 'weight': 3, 'content': [{'end': 2718.179, 'text': "You can try to figure this one out for yourself, and I think it's good practice to try to think about it.", 'start': 2711.614, 'duration': 6.565}, {'end': 2725.104, 'text': "But it's very, very difficult, at least compared to these three, okay? Order of magnitude more difficult.", 'start': 2718.639, 'duration': 6.465}, {'end': 2729.447, 'text': 'It turns out that the answer is n plus k minus 1 choose k.', 'start': 2725.524, 'duration': 3.923}, {'end': 2740.464, 'text': 'And for practice, you may want to try just choose some very small values of n and k and verify that this is correct in a small example.', 'start': 2731.741, 'duration': 8.723}], 'summary': 'The answer is n plus k minus 1 choose k, which is very difficult compared to other options.', 'duration': 28.85, 'max_score': 2711.614, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw2711614.jpg'}], 'start': 2135.718, 'title': 'Combinations, binomial coefficients, and full house probability', 'summary': 'Introduces binomial coefficients (n choose k) for choosing subsets, exemplifies with 52 choose 5 in poker. it also discusses the probability of a full house using counting methods and provides insights into the n choose k binomial coefficient and the n plus k minus 1 choose k result.', 'chapters': [{'end': 2331.72, 'start': 2135.718, 'title': 'Probability: combinations and binomial coefficients', 'summary': "Introduces the concept of binomial coefficients, denoted as n choose k, to represent the number of ways to choose a subset of size k from a group of n objects, where order doesn't matter, as exemplified by 52 choose 5 in the context of poker hands.", 'duration': 196.002, 'highlights': ["Binomial coefficient n choose k is defined as n! / ((n-k)! * k!), representing the number of ways to choose a subset of size k from a group of n objects, where order doesn't matter.", 'The notation 52C5 (52 choose 5) is used to represent the number of possible poker hands, calculated as 52! / ((52-5)! * 5!), providing a concrete example of binomial coefficients in the context of probability.', 'The explanation of the justification for n choose k follows almost immediately from the multiplication rule, illustrating the reasoning behind the formula n! / ((n-k)! * k!) through a step-by-step selection process.']}, {'end': 2788.585, 'start': 2332.02, 'title': 'Probability of full house & counting methods', 'summary': 'Discusses the probability of a full house in a deck of cards, using a specific example of three sevens and two tens, and explains the concept of sampling with and without replacement and the importance of order in counting methods, providing insights into the n choose k binomial coefficient and the n plus k minus 1 choose k result.', 'duration': 456.565, 'highlights': ['The probability of a full house is determined by the combination of choosing three cards of one rank and two of another from a deck of 52 cards, with 13 possibilities for the first rank and 12 possibilities for the second rank, resulting in a probability calculation using the multiplication rule and binomial coefficients.', 'The chapter introduces the concept of sampling with and without replacement and the significance of order in counting methods, outlining a two by two table to analyze the different possibilities and the immediate calculations for each case, emphasizing the importance of understanding the multiplication rule and the n choose k binomial coefficient.', 'The discussion delves into the n plus k minus 1 choose k result, highlighting its subtle complexity compared to other counting methods and encouraging further exploration and verification of the result through small examples or proofs, while emphasizing its usefulness in various applications and upcoming homework.', 'The chapter concludes by summarizing the covered concepts as essential for counting and homework, with the exception of a few additional topics to be addressed in the next session due to a holiday, urging the audience to commence their assignments and wishing them a good weekend.']}], 'duration': 652.867, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/KbB0FjPg0mw/pics/KbB0FjPg0mw2135718.jpg', 'highlights': ['The notation 52C5 (52 choose 5) exemplifies binomial coefficients in poker.', 'Binomial coefficient n choose k is defined as n! / ((n-k)! * k!), representing the number of ways to choose a subset of size k from a group of n objects.', 'The probability of a full house is determined by the combination of choosing three cards of one rank and two of another from a deck of 52 cards.', 'The discussion delves into the n plus k minus 1 choose k result, highlighting its subtle complexity compared to other counting methods.', 'The chapter introduces the concept of sampling with and without replacement and the significance of order in counting methods.']}], 'highlights': ["Probability's wide applications in various fields, including physics, genetics, history, finance, and gambling.", 'The significance of probability and statistics in quantifying uncertainty.', 'The historical significance of probability and its progress over time.', 'The importance of understanding events as subsets of the sample space.', 'The concept of sample space as the set of all possible outcomes.', 'The significance of set theory in enabling mathematical treatment of probability.', 'The importance of pattern recognition and extensive practice.', 'Strategic practice involves themed problem grouping for improved pattern recognition.', 'The significance of clarity and detailed reasoning in mathematical justifications.', 'Using English to explain equations for clear and detailed justifications.', 'The analogy of hiring someone to build a bridge illustrates the importance of clear reasoning.', 'The breakthrough in probability was to start thinking of events as subsets, emphasizing the need for mathematical precision.', 'Illustration of 36 possible outcomes when rolling two six-sided dice, emphasizing the concept of events as subsets.', 'The importance of working hard and thinking hard to overcome counterintuitive aspects in probability.', 'The naive definition of probability assumes all outcomes are equally likely and requires a finite sample space, which is a very strong assumption in many cases.', 'An example of a fair coin toss is used to illustrate the naive definition, where the probability is calculated by counting the favorable outcomes and dividing by the total possible outcomes.', 'The discussion highlights the limitations of the naive definition, pointing out that it may lead to absurd conclusions when applied to complex scenarios.', 'The multiplication rule states n1 times, n2 times, blah, blah, blah times nr overall possible outcomes for combined experiments, illustrated through a simple ice cream example with 6 possibilities.', 'Exponential growth is demonstrated through the example of 2 to the 10th power resulting in over 1,024 possibilities, emphasizing the rapid expansion of possibilities in sequential choices.', 'The definition of probability based on equally likely outcomes is inadequate for assessing probabilities in scenarios with unequal likelihoods.', 'The need for a more sophisticated approach to probability becomes evident when considering scenarios where strict inequalities in likelihood exist.', 'The notation 52C5 (52 choose 5) exemplifies binomial coefficients in poker.', 'Binomial coefficient n choose k is defined as n! / ((n-k)! * k!), representing the number of ways to choose a subset of size k from a group of n objects.', 'The probability of a full house is determined by the combination of choosing three cards of one rank and two of another from a deck of 52 cards.', 'The discussion delves into the n plus k minus 1 choose k result, highlighting its subtle complexity compared to other counting methods.', 'The chapter introduces the concept of sampling with and without replacement and the significance of order in counting methods.']}