title

Lecture 7: Gambler's Ruin and Random Variables | Statistics 110

description

We analyze the gambler's ruin problem, in which two gamblers bet with each other until one goes broke. We then introduce random variables, which are essential in statistics and for the rest of the course, and start on the Bernoulli and Binomial distributions.

detail

{'title': "Lecture 7: Gambler's Ruin and Random Variables | Statistics 110", 'heatmap': [{'end': 1931.693, 'start': 1893.894, 'weight': 1}], 'summary': "Lecture covers conditional probability, random variables, gambler's ruin problem, recursion, difference equations, and binomial distribution, providing insights into their significance in statistics with an emphasis on conditional thinking and analyzing the probability of winning in a game based on initial conditions and random walk concepts.", 'chapters': [{'end': 133.805, 'segs': [{'end': 25.263, 'src': 'embed', 'start': 0.189, 'weight': 0, 'content': [{'end': 8.576, 'text': "So we've been talking a lot about conditional probability, right? And we'll keep talking about conditional stuff for the entire semester.", 'start': 0.189, 'duration': 8.387}, {'end': 19.742, 'text': "So the plan for today is Do one more famous important example of thinking conditionally, and then we'll start on random variables.", 'start': 9.357, 'duration': 10.385}, {'end': 25.263, 'text': 'And if I had to name the two most important ideas for the entire semester,', 'start': 20.442, 'duration': 4.821}], 'summary': 'Discussing conditional probability and transitioning to random variables in the semester.', 'duration': 25.074, 'max_score': 0.189, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4189.jpg'}, {'end': 133.805, 'src': 'embed', 'start': 106.868, 'weight': 1, 'content': [{'end': 110.13, 'text': 'The other one is random variables and their distributions.', 'start': 106.868, 'duration': 3.262}, {'end': 116.053, 'text': "So that's what we're gonna start on in the second half today.", 'start': 112.311, 'duration': 3.742}, {'end': 122.616, 'text': "And then we'll be continuing doing random variables for the next few lectures, but also we'll be using them for the entire semester.", 'start': 116.693, 'duration': 5.923}, {'end': 133.805, 'text': 'And okay, I have had one stat 110 alumni has been on the Colbert report, but I think only one so far.', 'start': 125.738, 'duration': 8.067}], 'summary': 'Starting random variables, using them for entire semester. one stat 110 alumni on colbert report.', 'duration': 26.937, 'max_score': 106.868, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4106868.jpg'}], 'start': 0.189, 'title': 'Conditional probability and random variables', 'summary': 'Delves into the significance of conditioning and random variables in statistics, focusing on their distributions and emphasizing their importance throughout the semester.', 'chapters': [{'end': 133.805, 'start': 0.189, 'title': 'Conditional probability and random variables', 'summary': 'Discusses the importance of conditioning and random variables in statistics, with a focus on their distributions, as well as the upcoming topics on random variables, emphasizing their significance throughout the semester.', 'duration': 133.616, 'highlights': ['Conditioning and random variables are the two most important ideas for the entire semester, with conditioning being the soul of statistics.', 'The chapter emphasizes the significance of conditioning and random variables in statistics, including their distributions, and their continuous relevance throughout the semester.', 'The plan for today is to cover a famous important example of thinking conditionally and then start on random variables, setting the stage for the upcoming lectures.']}], 'duration': 133.616, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4189.jpg', 'highlights': ['Conditioning and random variables are the two most important ideas for the entire semester, with conditioning being the soul of statistics.', 'The chapter emphasizes the significance of conditioning and random variables in statistics, including their distributions, and their continuous relevance throughout the semester.', 'The plan for today is to cover a famous important example of thinking conditionally and then start on random variables, setting the stage for the upcoming lectures.']}, {'end': 639.899, 'segs': [{'end': 186.552, 'src': 'embed', 'start': 155.719, 'weight': 0, 'content': [{'end': 156.719, 'text': 'what should you condition on?', 'start': 155.719, 'duration': 1}, {'end': 158.62, 'text': 'how can you use it to solve problems?', 'start': 156.719, 'duration': 1.901}, {'end': 165.684, 'text': "So here's another very, very famous useful probability problem that's known as the gambler's ruin problem.", 'start': 158.7, 'duration': 6.984}, {'end': 172.607, 'text': "So here's the problem.", 'start': 171.747, 'duration': 0.86}, {'end': 178.502, 'text': "There are two gamblers, let's just call them Gambler A and Gambler B.", 'start': 174.837, 'duration': 3.665}, {'end': 182.727, 'text': "And they're just basically betting dollars back and forth.", 'start': 178.502, 'duration': 4.225}, {'end': 186.552, 'text': "So they're playing a sequence of the same game over and over and over again.", 'start': 182.787, 'duration': 3.765}], 'summary': 'Problem solving using conditional probability in gambling scenarios.', 'duration': 30.833, 'max_score': 155.719, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4155719.jpg'}, {'end': 264.927, 'src': 'embed', 'start': 227.46, 'weight': 7, 'content': [{'end': 228.641, 'text': "okay?. So that's the setup.", 'start': 227.46, 'duration': 1.181}, {'end': 236.589, 'text': 'So two gamblers, A and B, with the sequence.', 'start': 229.822, 'duration': 6.767}, {'end': 245.963, 'text': "I'll call them, to distinguish, I'm thinking of the word game as the entire process, and each round is one individual bet.", 'start': 238.922, 'duration': 7.041}, {'end': 251.284, 'text': 'Sequence of rounds where they bet $1 each time.', 'start': 246.503, 'duration': 4.781}, {'end': 257.204, 'text': "Each round is independent, they're just a sequence of independent bets, $1 each time.", 'start': 251.704, 'duration': 5.5}, {'end': 264.927, 'text': "And let's assume that P equals the probability that A wins.", 'start': 258.166, 'duration': 6.761}], 'summary': "Two gamblers, a and b, bet $1 each round with p as a's winning probability.", 'duration': 37.467, 'max_score': 227.46, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4227460.jpg'}, {'end': 329.692, 'src': 'embed', 'start': 294.851, 'weight': 1, 'content': [{'end': 302.541, 'text': "And the problem is, is to find what's the probability that A wins the entire game.", 'start': 294.851, 'duration': 7.69}, {'end': 308.822, 'text': "That is, that A wins all the money and B goes bankrupt, okay? So that's the problem.", 'start': 302.661, 'duration': 6.161}, {'end': 317.945, 'text': 'So find the probability that A wins the entire game.', 'start': 310.223, 'duration': 7.722}, {'end': 325.888, 'text': 'So B is ruined, we would say.', 'start': 322.465, 'duration': 3.423}, {'end': 329.692, 'text': "Or we could ask the other question what's the probability that A gets ruined?", 'start': 326.529, 'duration': 3.163}], 'summary': 'Find the probability of a winning the entire game and b going bankrupt.', 'duration': 34.841, 'max_score': 294.851, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4294851.jpg'}, {'end': 397.868, 'src': 'embed', 'start': 371.291, 'weight': 3, 'content': [{'end': 378.393, 'text': "I chose the notation this way just so that I mean I just makes me think of an integer, because I'm assuming integer dollars going back and forth.", 'start': 371.291, 'duration': 7.102}, {'end': 385.463, 'text': "And I call B's initial wealth N-i because then the total is capital N dollars.", 'start': 379.38, 'duration': 6.083}, {'end': 389.504, 'text': 'So this is a closed system, right? No money is coming in or going out.', 'start': 385.843, 'duration': 3.661}, {'end': 397.868, 'text': "So there's basically N dollars on the table that are just going back and forth, back and forth until one player has all of the capital N dollars.", 'start': 390.165, 'duration': 7.703}], 'summary': 'Closed system with n dollars going back and forth until one player has all n dollars.', 'duration': 26.577, 'max_score': 371.291, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4371291.jpg'}, {'end': 469.511, 'src': 'embed', 'start': 441.403, 'weight': 2, 'content': [{'end': 446.524, 'text': 'So one of the most important things in this course is just being able to recognize pattern right?', 'start': 441.403, 'duration': 5.121}, {'end': 450.626, 'text': "So we may see problems later that it's really a gambler's ruin.", 'start': 447.285, 'duration': 3.341}, {'end': 455.627, 'text': "It's not phrased in terms of two gamblers, A and B, but the problem has the same structure.", 'start': 450.666, 'duration': 4.961}, {'end': 462.829, 'text': "So an example of a problem that's exactly the same as this would be if we had a random walk that we can picture like this.", 'start': 456.247, 'duration': 6.582}, {'end': 464.79, 'text': 'Here, just draw a number line.', 'start': 463.409, 'duration': 1.381}, {'end': 469.511, 'text': "And here's 0, and here's capital N, and here's I.", 'start': 466.01, 'duration': 3.501}], 'summary': 'Recognizing patterns is crucial in this course, with examples of problems showing similar structures.', 'duration': 28.108, 'max_score': 441.403, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4441403.jpg'}, {'end': 615.734, 'src': 'embed', 'start': 544.123, 'weight': 4, 'content': [{'end': 546.345, 'text': 'So we would call that an absorbing state.', 'start': 544.123, 'duration': 2.222}, {'end': 551.669, 'text': 'So we have absorbing states at 0 and n.', 'start': 547.325, 'duration': 4.344}, {'end': 558.994, 'text': 'Just think of that as like a black hole or some very sticky place where it just gets stuck there and then the game is over.', 'start': 551.669, 'duration': 7.325}, {'end': 560.015, 'text': 'It just stays there forever.', 'start': 559.014, 'duration': 1.001}, {'end': 561.055, 'text': 'So it gets trapped.', 'start': 560.295, 'duration': 0.76}, {'end': 563.037, 'text': "So that's called an absorbing state.", 'start': 561.676, 'duration': 1.361}, {'end': 568.573, 'text': "then the game's over, that's the same thing as here.", 'start': 564.688, 'duration': 3.885}, {'end': 574.68, 'text': 'So think of this random walk as just tracking how much money A has.', 'start': 571.096, 'duration': 3.584}, {'end': 581.028, 'text': 'Of course you could draw an analogous random walk for B, but we may as well just track A and then B is determined, right?', 'start': 575.141, 'duration': 5.887}, {'end': 586.171, 'text': 'So if A ever ends up here, It means A has all the money and B is bankrupt.', 'start': 581.629, 'duration': 4.542}, {'end': 589.736, 'text': 'And if A ends up here, then A is bankrupt and B has all the money, okay?', 'start': 586.211, 'duration': 3.525}, {'end': 594.562, 'text': "So it's the exact same problem, just in a different way to picture it, different interpretation.", 'start': 589.756, 'duration': 4.806}, {'end': 596.986, 'text': "Okay, so that's the problem.", 'start': 595.904, 'duration': 1.082}, {'end': 598.648, 'text': "Now let's try to solve the problem.", 'start': 597.306, 'duration': 1.342}, {'end': 610.711, 'text': "Well, at first, This may seem like it's not so easy, right? I mean, you don't know how many rounds the game will last.", 'start': 599.709, 'duration': 11.002}, {'end': 612.673, 'text': "You don't really know.", 'start': 611.272, 'duration': 1.401}, {'end': 615.734, 'text': 'is it possible that this would wander forever?', 'start': 612.673, 'duration': 3.061}], 'summary': 'Random walk models absorbing states and financial outcomes in a game scenario.', 'duration': 71.611, 'max_score': 544.123, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4544123.jpg'}], 'start': 133.905, 'title': "Gambler's ruin problem", 'summary': "Introduces the famous gambler's ruin problem, analyzing the probability of a gambler winning a game by bankrupting the opponent, based on initial conditions and random walk concepts.", 'chapters': [{'end': 412.061, 'start': 133.905, 'title': "Gambler's ruin probability problem", 'summary': "Introduces the famous gambler's ruin problem, where two gamblers, a and b, play a sequence of rounds betting $1 each time, and the goal is to find the probability of a winning the entire game by bankrupting b, given the initial conditions of a starting with i dollars and b starting with n - i dollars.", 'duration': 278.156, 'highlights': ["The chapter introduces the famous gambler's ruin problem", 'Two gamblers, A and B, play a sequence of rounds betting $1 each time', 'The goal is to find the probability of A winning the entire game by bankrupting B', 'Given the initial conditions of A starting with I dollars and B starting with N - I dollars']}, {'end': 639.899, 'start': 412.161, 'title': "Random walk and gambler's ruin", 'summary': "Introduces the concept of a random walk and its application to the gambler's ruin problem, where a random walk with absorbing states represents the scenario of two gamblers, a and b, in a financial context.", 'duration': 227.738, 'highlights': ["The chapter introduces the concept of a random walk and its application to the gambler's ruin problem.", 'The scenario of two gamblers, A and B, is represented by a random walk with absorbing states in a financial context.', 'The concept of absorbing states in the context of a random walk is explained, representing the scenario in which the game ends.', "The uncertainty of the game's duration and the possibility of the random walk wandering forever is discussed."]}], 'duration': 505.994, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4133905.jpg', 'highlights': ["The chapter introduces the famous gambler's ruin problem", 'The goal is to find the probability of A winning the entire game by bankrupting B', "The chapter introduces the concept of a random walk and its application to the gambler's ruin problem", 'Given the initial conditions of A starting with I dollars and B starting with N - I dollars', 'The scenario of two gamblers, A and B, is represented by a random walk with absorbing states in a financial context', 'The concept of absorbing states in the context of a random walk is explained, representing the scenario in which the game ends', "The uncertainty of the game's duration and the possibility of the random walk wandering forever is discussed", 'Two gamblers, A and B, play a sequence of rounds betting $1 each time']}, {'end': 937.411, 'segs': [{'end': 715.821, 'src': 'embed', 'start': 688.906, 'weight': 0, 'content': [{'end': 695.849, 'text': 'Well, the general strategy is we have a complicated problem, we find something to condition on to make it simpler.', 'start': 688.906, 'duration': 6.943}, {'end': 700.551, 'text': 'A more specific strategy here is condition on the first step.', 'start': 696.289, 'duration': 4.262}, {'end': 704.394, 'text': 'So sometimes this is called first step analysis.', 'start': 702.113, 'duration': 2.281}, {'end': 712.319, 'text': "There are a lot of problems where that's something that's useful to condition to.", 'start': 708.497, 'duration': 3.822}, {'end': 715.821, 'text': 'Remember, we were talking about wishful thinking like what do we wish we knew?', 'start': 712.339, 'duration': 3.482}], 'summary': 'Strategy: simplify by conditioning on first step. useful in many problems.', 'duration': 26.915, 'max_score': 688.906, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4688906.jpg'}, {'end': 765.303, 'src': 'embed', 'start': 733.683, 'weight': 1, 'content': [{'end': 740.067, 'text': "maybe you condition on the first two rounds or the first three rounds, but don't wanna condition on the entire future of the universe.", 'start': 733.683, 'duration': 6.384}, {'end': 743.587, 'text': "Okay, so let's try conditioning on the first step.", 'start': 741.365, 'duration': 2.222}, {'end': 745.228, 'text': "So that's called first step analysis.", 'start': 743.727, 'duration': 1.501}, {'end': 750.432, 'text': "And then let's write down what's the equation.", 'start': 746.729, 'duration': 3.703}, {'end': 752.053, 'text': 'So we need some notation.', 'start': 750.872, 'duration': 1.181}, {'end': 765.303, 'text': 'So let p sub i equal the probability that A wins the entire game, not just a round, given that A starts with i dollars.', 'start': 752.093, 'duration': 13.21}], 'summary': 'Using first step analysis to calculate probability of a winning the game.', 'duration': 31.62, 'max_score': 733.683, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4733683.jpg'}, {'end': 937.411, 'src': 'embed', 'start': 890.652, 'weight': 3, 'content': [{'end': 900.743, 'text': "Okay, so as far as we're concerned in this course, this is the most important thing about this problem.", 'start': 890.652, 'duration': 10.091}, {'end': 904.807, 'text': 'Well, the two most important things are one is just recognizing the problem.', 'start': 901.623, 'duration': 3.184}, {'end': 908.25, 'text': "If we had another problem that was the same as the gambler's ruin, recognizing the structure.", 'start': 904.927, 'duration': 3.323}, {'end': 911.734, 'text': 'And the other is thinking conditionally, write this thing down.', 'start': 908.631, 'duration': 3.103}, {'end': 914.478, 'text': 'Notice this is a recursive equation.', 'start': 912.774, 'duration': 1.704}, {'end': 918.545, 'text': 'So at this point, if you wanted to solve it either kind of tediously,', 'start': 914.538, 'duration': 4.007}, {'end': 923.074, 'text': 'by hand or on a computer computers are very good at recursion this would be very easy.', 'start': 918.545, 'duration': 4.529}, {'end': 931.628, 'text': "Because we could rewrite this for, well we can rearrange, at least once we have, actually it's a little bit more complicated.", 'start': 925.126, 'duration': 6.502}, {'end': 937.411, 'text': 'Cuz usually with a recursion we get the first initial conditions and then we can solve from there.', 'start': 931.949, 'duration': 5.462}], 'summary': 'Recognizing and solving recursive equations is crucial for problem-solving in this course.', 'duration': 46.759, 'max_score': 890.652, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4890652.jpg'}], 'start': 639.919, 'title': "Recursion and gambler's ruin", 'summary': "Discusses the recursive structure of problems, introduces first step analysis, and explores the gambler's ruin problem, emphasizing conditional thinking and providing insights into solving complex problems.", 'chapters': [{'end': 752.053, 'start': 639.919, 'title': 'Recursion and first step analysis', 'summary': 'Discusses the recursive structure of the problem and introduces the strategy of first step analysis to simplify the complicated problem by conditioning on the outcome of the first round, providing insight into breaking it into simpler, manageable pieces.', 'duration': 112.134, 'highlights': ['The problem has a recursive structure, where it becomes the same problem with a different initial condition after one step, leading to the strategy of first step analysis. (Relevance: 5)', 'The specific strategy introduced is to condition on the outcome of the first round, termed as first step analysis, to simplify the complicated problem. (Relevance: 4)', 'The chapter emphasizes the need to find something to condition on in order to break the problem into simpler, manageable pieces and avoid getting too greedy with the conditioning. (Relevance: 3)']}, {'end': 937.411, 'start': 752.093, 'title': "Gambler's ruin problem", 'summary': "Discusses the probability of a winning the entire game given different starting amounts, emphasizing the importance of recognizing the problem's structure and thinking conditionally, and deriving a recursive equation for solving it.", 'duration': 185.318, 'highlights': ["The importance of recognizing the problem's structure and thinking conditionally.", 'Deriving a recursive equation for solving the problem.', 'The equation for probability pi and its immediate derivation from the law of total probability.']}], 'duration': 297.492, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4639919.jpg', 'highlights': ['The problem has a recursive structure, leading to first step analysis. (Relevance: 5)', 'The specific strategy is to condition on the outcome of the first round. (Relevance: 4)', 'The need to find something to condition on to simplify the problem. (Relevance: 3)', 'Deriving a recursive equation for solving the problem. (Relevance: 2)', "Importance of recognizing the problem's structure and thinking conditionally. (Relevance: 1)"]}, {'end': 1990.353, 'segs': [{'end': 1025.865, 'src': 'embed', 'start': 938.351, 'weight': 3, 'content': [{'end': 947.074, 'text': "Here, let's see, we can rearrange this for pi plus 1 in terms of pi and pi minus 1.", 'start': 938.351, 'duration': 8.723}, {'end': 951.916, 'text': 'So it has this recursive flavor where terms are defined in terms of other terms.', 'start': 947.074, 'duration': 4.842}, {'end': 958.235, 'text': 'Okay, so this is called a difference equation, not to be confused with differential equation.', 'start': 953.174, 'duration': 5.061}, {'end': 967.177, 'text': 'A difference equation is the discrete analog of a differential equation.', 'start': 962.296, 'duration': 4.881}, {'end': 976.139, 'text': 'And one thing that kind of annoys me is that difference equations never get taught, as far as I know, anywhere.', 'start': 967.857, 'duration': 8.282}, {'end': 977.299, 'text': "That's pretty sad.", 'start': 976.639, 'duration': 0.66}, {'end': 985.263, 'text': 'My opinion is that difference equations are at least as important as differential equations.', 'start': 979.32, 'duration': 5.943}, {'end': 988.204, 'text': 'Differential equations get taught all over the place.', 'start': 985.283, 'duration': 2.921}, {'end': 992.806, 'text': "It's the continuous version, and difference equations just get really neglected.", 'start': 989.465, 'duration': 3.341}, {'end': 1000.43, 'text': 'And the saddest thing was, I looked up difference equation on Google yesterday.', 'start': 993.267, 'duration': 7.163}, {'end': 1005.192, 'text': "The number one link is Wikipedia's entry on differential equations.", 'start': 1000.45, 'duration': 4.742}, {'end': 1009.006, 'text': 'So that was really annoying.', 'start': 1007.924, 'duration': 1.082}, {'end': 1013.091, 'text': 'Does anyone know if any math class here teaches how to solve difference equations?', 'start': 1009.366, 'duration': 3.725}, {'end': 1024.364, 'text': 'Okay, so, anyway, these equations come up everywhere right?', 'start': 1017.901, 'duration': 6.463}, {'end': 1025.865, 'text': "In fact, they're more realistic.", 'start': 1024.645, 'duration': 1.22}], 'summary': 'Difference equations are important but neglected, with little teaching and recognition compared to differential equations.', 'duration': 87.514, 'max_score': 938.351, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4938351.jpg'}, {'end': 1101.517, 'src': 'embed', 'start': 1075.339, 'weight': 0, 'content': [{'end': 1079.326, 'text': 'This is actually, I mean I could just state the general theory of difference equations,', 'start': 1075.339, 'duration': 3.987}, {'end': 1085.306, 'text': "but I think it's easier to just see what happens For this example.", 'start': 1079.326, 'duration': 5.98}, {'end': 1086.167, 'text': 'how would we solve it?', 'start': 1085.306, 'duration': 0.861}, {'end': 1093.291, 'text': 'And then, actually, what I will have shown you will help you to solve any difference equation that you encounter.', 'start': 1086.267, 'duration': 7.024}, {'end': 1099.375, 'text': "Anyway, at least if it's of this form where the coefficients are constants and it's what we call linear.", 'start': 1093.611, 'duration': 5.764}, {'end': 1101.517, 'text': "Okay, so let's try to solve this equation.", 'start': 1099.896, 'duration': 1.621}], 'summary': 'Demonstrating how to solve a linear difference equation with constant coefficients.', 'duration': 26.178, 'max_score': 1075.339, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41075339.jpg'}, {'end': 1386.324, 'src': 'embed', 'start': 1355.112, 'weight': 4, 'content': [{'end': 1360.777, 'text': 'So, for those of you who studied differential equations, first you try to solve the general equation right?', 'start': 1355.112, 'duration': 5.665}, {'end': 1364.48, 'text': 'And then you have some initial or boundary conditions and you get a specific solution.', 'start': 1360.797, 'duration': 3.683}, {'end': 1379.382, 'text': 'Okay. so the general solution that we should look at is gonna be a times 1 to the pi equals a times 1 to the i plus b,', 'start': 1365.879, 'duration': 13.503}, {'end': 1386.324, 'text': 'times q over p to the i if p not equal q.', 'start': 1379.382, 'duration': 6.942}], 'summary': 'In differential equations, the general solution is a times 1 to the pi equals a times 1 to the i plus b times q over p to the i if p not equal q.', 'duration': 31.212, 'max_score': 1355.112, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41355112.jpg'}, {'end': 1635.152, 'src': 'embed', 'start': 1610.713, 'weight': 2, 'content': [{'end': 1616.714, 'text': 'Now, the x stuff is just going to 1, so we just have i over n.', 'start': 1610.713, 'duration': 6.001}, {'end': 1619.015, 'text': 'So I claim that the answer here is i over n.', 'start': 1616.714, 'duration': 2.301}, {'end': 1628.546, 'text': "Now, it could be that something very shocking happens and the limit of this solution isn't this solution.", 'start': 1621.38, 'duration': 7.166}, {'end': 1635.152, 'text': 'Intuitively, it should be true that if you let the game get more and more fair, this is the unfair game solution.', 'start': 1629.827, 'duration': 5.325}], 'summary': 'The answer is i over n, representing the unfair game solution.', 'duration': 24.439, 'max_score': 1610.713, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41610713.jpg'}, {'end': 1733.402, 'src': 'embed', 'start': 1708.747, 'weight': 1, 'content': [{'end': 1714.951, 'text': 'the probability that A wins is proportional actually just is equal to what fraction of the wealth that A has.', 'start': 1708.747, 'duration': 6.204}, {'end': 1718.993, 'text': 'So if A starts out with two-thirds of the money, A has a two-thirds chance of winning.', 'start': 1715.011, 'duration': 3.982}, {'end': 1721.435, 'text': "So that's easy to remember and that's pretty neat.", 'start': 1719.413, 'duration': 2.022}, {'end': 1729.44, 'text': "And let's see, I did just numerically a few calculations for what happens in the unfair case.", 'start': 1722.896, 'duration': 6.544}, {'end': 1733.402, 'text': 'So this is the fair case I think is then pretty intuitive if you think of it the way I just said.', 'start': 1729.76, 'duration': 3.642}], 'summary': "Probability of a winning is based on a's wealth fraction, with 2/3 chance if a starts with 2/3 money.", 'duration': 24.655, 'max_score': 1708.747, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41708747.jpg'}, {'end': 1785.266, 'src': 'embed', 'start': 1751.001, 'weight': 5, 'content': [{'end': 1760.107, 'text': 'So the game is just very slightly unfair towards gambler A, right? Just only by 0.01 away from being fair.', 'start': 1751.001, 'duration': 9.106}, {'end': 1762.649, 'text': 'Slightly unfair each round.', 'start': 1760.787, 'duration': 1.862}, {'end': 1764.43, 'text': 'Okay, and then see what happens.', 'start': 1763.169, 'duration': 1.261}, {'end': 1773.418, 'text': 'Well, if n equals 20, then the chance that A wins is 40%.', 'start': 1766.072, 'duration': 7.346}, {'end': 1775.999, 'text': "So that doesn't sound too bad.", 'start': 1773.418, 'duration': 2.581}, {'end': 1785.266, 'text': "If n is 100, so that's the case where each player starts out with $50 and the game is just a tiny bit unfavorable to player A,", 'start': 1776.64, 'duration': 8.626}], 'summary': "Game slightly unfair, a's chance to win is 40% with n=20 and game unfavorable to a with n=100.", 'duration': 34.265, 'max_score': 1751.001, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41751001.jpg'}, {'end': 1931.693, 'src': 'heatmap', 'start': 1893.894, 'weight': 1, 'content': [{'end': 1895.835, 'text': "What's the probability that B wins and A is ruined?", 'start': 1893.894, 'duration': 1.941}, {'end': 1899.976, 'text': "Well, we're not really gonna go through this entire thing again, right?", 'start': 1896.815, 'duration': 3.161}, {'end': 1905.377, 'text': "It's just like all we did was change the name of player A to B and change the name of B to A.", 'start': 1900.416, 'duration': 4.961}, {'end': 1907.837, 'text': 'we can immediately write down the answer, okay?', 'start': 1905.377, 'duration': 2.46}, {'end': 1913.038, 'text': 'So from this we can immediately write down the probability that B wins and A is ruined.', 'start': 1908.157, 'duration': 4.881}, {'end': 1918.82, 'text': "Now, if you do that, and if you add those two numbers, let's just do it quickly for this case.", 'start': 1913.619, 'duration': 5.201}, {'end': 1925.681, 'text': 'If we did the same thing for B, we would get n-i over n, because n-i was how much B started with.', 'start': 1920.12, 'duration': 5.561}, {'end': 1931.693, 'text': "If we add, so I'm just gonna mention this for the fair case, but the same thing happens in the unfair case.", 'start': 1926.229, 'duration': 5.464}], 'summary': 'Probability of b winning and a being ruined is calculated by changing player names, yielding n-i over n.', 'duration': 37.799, 'max_score': 1893.894, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41893894.jpg'}, {'end': 1990.353, 'src': 'embed', 'start': 1960.553, 'weight': 7, 'content': [{'end': 1965.234, 'text': 'But probabilistically and statistically it has probably 0, so it would never actually happen.', 'start': 1960.553, 'duration': 4.681}, {'end': 1969.496, 'text': 'Probably 0 of game going on forever.', 'start': 1967.175, 'duration': 2.321}, {'end': 1979.319, 'text': "So we didn't have to do a separate calculation of what's the probability that the game ends, because the probability that A wins,", 'start': 1973.977, 'duration': 5.342}, {'end': 1981.82, 'text': "plus the probability that B wins, is 1, there's nothing left over.", 'start': 1979.319, 'duration': 2.501}, {'end': 1990.353, 'text': "All right, so that's the gambler's ruin problem.", 'start': 1982.93, 'duration': 7.423}], 'summary': "Probability of game going on forever is close to 0, leading to the gambler's ruin problem.", 'duration': 29.8, 'max_score': 1960.553, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41960553.jpg'}], 'start': 938.351, 'title': "Difference equations and gambler's ruin", 'summary': "Discusses the importance of difference equations and their neglect in education, as well as solving a specific difference equation and exploring the implications, and explains the concept of the gambler's ruin with numerical examples showcasing the changing probabilities and inevitability of losing in an unfair game.", 'chapters': [{'end': 1025.865, 'start': 938.351, 'title': 'Importance of difference equations', 'summary': 'Discusses the importance of difference equations, stating that they are as important as differential equations but are often neglected in education, despite being more realistic and coming up everywhere.', 'duration': 87.514, 'highlights': ['Difference equations are defined in terms of other terms, exhibiting a recursive flavor and being the discrete analog of a differential equation.', 'The speaker expresses frustration over the neglect of difference equations in education, emphasizing their importance and prevalence in various fields.', "The top Google link for 'difference equation' leads to Wikipedia's entry on differential equations, indicating the lack of prominence given to difference equations in educational resources."]}, {'end': 1653.472, 'start': 1027.086, 'title': 'Solving difference equations', 'summary': 'Discusses solving a specific difference equation through a unique approach, resulting in an explicit solution pi = 1 - q/p to the i / 1 - q/p to the n, and explores the implications of p=q on the solution, concluding with a limit as x goes to 1 yielding i/n.', 'duration': 626.386, 'highlights': ['The chapter presents a unique approach to solving a specific difference equation, resulting in the explicit solution pi = 1 - q/p to the i / 1 - q/p to the n, providing a direct analytical solution to the problem.', "The implications of p=q on the solution are explored, leading to a discussion of the game's fairness and the consideration of a limit as x goes to 1, yielding the result i/n, indicating the convergence of the unfair game solution to the fair solution as the game becomes fairer.", "The chapter emphasizes the significance of the solution, suggesting its general applicability to any similar difference equation where the coefficients are constants and it's a linear form, providing a versatile and broadly applicable solution method."]}, {'end': 1990.353, 'start': 1653.512, 'title': "Understanding the gambler's ruin", 'summary': "Explains the concept of the gambler's ruin, demonstrating how the probability of winning in a gambling scenario changes based on the initial wealth and the fairness of the game, with specific numerical examples highlighting the increasing disadvantage for a gambler in an unfair game and the inevitability of losing all money with high probability.", 'duration': 336.841, 'highlights': ['The probability of winning in a fair gambling scenario is directly proportional to the fraction of wealth the player initially possesses, with specific numerical examples provided to illustrate this intuitive concept.', 'In an unfair gambling scenario where the game slightly favors one player, the probability of winning decreases significantly, with specific numerical examples showing the decreasing chances of winning as the game becomes increasingly unfair.', "The analysis of the gambler's ruin problem also reveals that the probability of the game oscillating forever is 0, indicating the inevitability of the game reaching a conclusion rather than continuing indefinitely."]}], 'duration': 1052.002, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi4938351.jpg', 'highlights': ["The chapter emphasizes the significance of the solution, suggesting its general applicability to any similar difference equation where the coefficients are constants and it's a linear form, providing a versatile and broadly applicable solution method.", 'The probability of winning in a fair gambling scenario is directly proportional to the fraction of wealth the player initially possesses, with specific numerical examples provided to illustrate this intuitive concept.', "The implications of p=q on the solution are explored, leading to a discussion of the game's fairness and the consideration of a limit as x goes to 1, yielding the result i/n, indicating the convergence of the unfair game solution to the fair solution as the game becomes fairer.", 'The speaker expresses frustration over the neglect of difference equations in education, emphasizing their importance and prevalence in various fields.', 'The chapter presents a unique approach to solving a specific difference equation, resulting in the explicit solution pi = 1 - q/p to the i / 1 - q/p to the n, providing a direct analytical solution to the problem.', 'In an unfair gambling scenario where the game slightly favors one player, the probability of winning decreases significantly, with specific numerical examples showing the decreasing chances of winning as the game becomes increasingly unfair.', 'Difference equations are defined in terms of other terms, exhibiting a recursive flavor and being the discrete analog of a differential equation.', "The analysis of the gambler's ruin problem also reveals that the probability of the game oscillating forever is 0, indicating the inevitability of the game reaching a conclusion rather than continuing indefinitely.", "The top Google link for 'difference equation' leads to Wikipedia's entry on differential equations, indicating the lack of prominence given to difference equations in educational resources."]}, {'end': 3104.646, 'segs': [{'end': 2015.632, 'src': 'embed', 'start': 1990.513, 'weight': 4, 'content': [{'end': 1996.695, 'text': "So spend the rest of today starting random variables and we'll just be continuing lots and lots and lots of random variable stuff.", 'start': 1990.513, 'duration': 6.182}, {'end': 2000.217, 'text': 'So why do we need random variables?', 'start': 1998.716, 'duration': 1.501}, {'end': 2010.981, 'text': "Well, you can already start to see that the notation starts to get kind of unwieldy when we're talking about this problem.", 'start': 2001.197, 'duration': 9.784}, {'end': 2015.632, 'text': "So far, probability is always, everything's defined in terms of events.", 'start': 2012.61, 'duration': 3.022}], 'summary': 'Introduction to random variables for probability analysis.', 'duration': 25.119, 'max_score': 1990.513, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41990513.jpg'}, {'end': 2420.798, 'src': 'embed', 'start': 2381.782, 'weight': 2, 'content': [{'end': 2383.142, 'text': 'so where does the randomness come from?', 'start': 2381.782, 'duration': 1.36}, {'end': 2398.499, 'text': "So the definition is that it is a function From the sample space, Which we've been calling S, that is,", 'start': 2383.162, 'duration': 15.337}, {'end': 2403.122, 'text': 'we have some random experiment to the real line.', 'start': 2398.499, 'duration': 4.623}, {'end': 2412.314, 'text': "So as input, We have this sample space and we've been drawing these Venn diagrams.", 'start': 2405.503, 'duration': 6.811}, {'end': 2420.798, 'text': 'The input is some possible outcome, little less is some, we have an experiment that has different possible outcomes.', 'start': 2413.075, 'duration': 7.723}], 'summary': 'Randomness comes from mapping sample space to the real line.', 'duration': 39.016, 'max_score': 2381.782, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi42381782.jpg'}, {'end': 2475.158, 'src': 'embed', 'start': 2444.984, 'weight': 5, 'content': [{'end': 2455.009, 'text': "And how can you reconcile this is the mathematical definition to try to reconcile this with this vague intuition that's supposed to be a random quantity,", 'start': 2444.984, 'duration': 10.025}, {'end': 2460.112, 'text': "right?. So that's something we'll talk a lot about, but the sooner you start thinking about this, the better.", 'start': 2455.009, 'duration': 5.103}, {'end': 2472.236, 'text': 'So the idea is that we should think of a random variable just as a numerical summary.', 'start': 2462.509, 'duration': 9.727}, {'end': 2475.158, 'text': 'And summary is interpreted very broadly.', 'start': 2473.156, 'duration': 2.002}], 'summary': 'Random variables are numerical summaries with broad interpretations, to be reconciled with mathematical definitions and vague intuition.', 'duration': 30.174, 'max_score': 2444.984, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi42444984.jpg'}, {'end': 2664.645, 'src': 'embed', 'start': 2622.523, 'weight': 3, 'content': [{'end': 2627.667, 'text': 'Right now, just think of that as the definition of the word Bernoulli, the term Bernoulli distribution.', 'start': 2622.523, 'duration': 5.144}, {'end': 2631.49, 'text': 'And often I abbreviate Bernoulli to Bern.', 'start': 2629.208, 'duration': 2.282}, {'end': 2648.723, 'text': 'If X has two possible values, only two possible values, 0 and 1.', 'start': 2632.356, 'duration': 16.367}, {'end': 2650.204, 'text': "And let's say.", 'start': 2648.723, 'duration': 1.481}, {'end': 2657.499, 'text': "I'm gonna call this Bernoulli p.", 'start': 2652.935, 'duration': 4.564}, {'end': 2664.645, 'text': 'If the probability that x equals 1 equals p, and the probability that x equals 0 equals 1- p.', 'start': 2657.499, 'duration': 7.146}], 'summary': 'Bernoulli distribution has two possible values (0 and 1) with probabilities p and 1-p.', 'duration': 42.122, 'max_score': 2622.523, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi42622523.jpg'}, {'end': 2863.891, 'src': 'embed', 'start': 2832.591, 'weight': 1, 'content': [{'end': 2840.335, 'text': 'The distribution of the number of successes in n independent Bernoulli trials is called the binomial np.', 'start': 2832.591, 'duration': 7.744}, {'end': 2858.57, 'text': "And its distribution is given by, to specify, a distribution just means, I'll define distribution more precisely next time.", 'start': 2844.177, 'duration': 14.393}, {'end': 2860.69, 'text': 'But intuitively, we have a random variable.', 'start': 2858.61, 'duration': 2.08}, {'end': 2863.891, 'text': 'The distribution is just kind of like the blueprint.', 'start': 2860.93, 'duration': 2.961}], 'summary': 'The binomial distribution describes the number of successes in n independent bernoulli trials.', 'duration': 31.3, 'max_score': 2832.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi42832591.jpg'}, {'end': 3059.08, 'src': 'embed', 'start': 3030.648, 'weight': 0, 'content': [{'end': 3034.69, 'text': "That is, we'll talk more later about what does it mean for two random variables to be independent formally.", 'start': 3030.648, 'duration': 4.042}, {'end': 3039.873, 'text': 'But intuitively, it just means that the events for X are completely independent of the events for Y.', 'start': 3034.75, 'duration': 5.123}, {'end': 3052.295, 'text': 'Then x plus y is binomial n plus m p.', 'start': 3041.368, 'duration': 10.927}, {'end': 3059.08, 'text': "And the proof, you could try to prove it, and we'll talk more about this later, how would you prove it in terms of these equations.", 'start': 3052.295, 'duration': 6.785}], 'summary': 'Independent random variables have events completely unrelated, leading to x+y being binomial n+m p.', 'duration': 28.432, 'max_score': 3030.648, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi43030648.jpg'}], 'start': 1990.513, 'title': 'Random variables and binomial distribution', 'summary': 'Introduces random variables, explaining their importance in probability problems, defining them as functions from sample space to the real line, and providing an example of the bernoulli distribution. it also covers the binomial distribution, including its probability mass function and the addition property of independent binomial random variables.', 'chapters': [{'end': 2171.407, 'start': 1990.513, 'title': 'Introduction to random variables', 'summary': 'Introduces the concept of random variables, highlighting the need for them due to unwieldy notation in probability problems and the vague and complex definitions of random variables.', 'duration': 180.894, 'highlights': ['The chapter explains the need for random variables due to unwieldy notation in probability problems.', 'The transcript criticizes the vague and complex definitions of random variables.']}, {'end': 2737.161, 'start': 2171.447, 'title': 'Understanding random variables', 'summary': 'Explains the concept of variables, the definition of a random variable as a function from the sample space to the real line, and provides an example of the bernoulli distribution and its notation.', 'duration': 565.714, 'highlights': ['The chapter explains the concept of variables and the definition of a random variable as a function from the sample space to the real line.', 'Provides an example of the Bernoulli distribution and its notation.']}, {'end': 3104.646, 'start': 2738.354, 'title': 'Binomial distribution basics', 'summary': 'Introduces the binomial distribution, which models the number of successes in n independent bernoulli trials, and explains its probability mass function and the addition property of independent binomial random variables.', 'duration': 366.292, 'highlights': ['The binomial distribution models the number of successes in n independent Bernoulli trials, with the probability mass function specified as p^k * (1-p)^(n-k) * n choose k, for k successes out of n trials.', 'The addition property states that if X is binomial NP and Y is binomial NP, and they are independent, then X + Y is binomial n+m p, where n and m are the numbers of trials for X and Y respectively.']}], 'duration': 1114.133, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/PNrqCdslGi4/pics/PNrqCdslGi41990513.jpg', 'highlights': ['The addition property states that if X is binomial NP and Y is binomial NP, and they are independent, then X + Y is binomial n+m p, where n and m are the numbers of trials for X and Y respectively.', 'The binomial distribution models the number of successes in n independent Bernoulli trials, with the probability mass function specified as p^k * (1-p)^(n-k) * n choose k, for k successes out of n trials.', 'The chapter explains the concept of variables and the definition of a random variable as a function from the sample space to the real line.', 'Provides an example of the Bernoulli distribution and its notation.', 'The chapter explains the need for random variables due to unwieldy notation in probability problems.', 'The transcript criticizes the vague and complex definitions of random variables.']}], 'highlights': ['Conditioning and random variables are the two most important ideas for the entire semester, with conditioning being the soul of statistics.', "The chapter introduces the famous gambler's ruin problem", 'The addition property states that if X is binomial NP and Y is binomial NP, and they are independent, then X + Y is binomial n+m p, where n and m are the numbers of trials for X and Y respectively.', "The chapter emphasizes the significance of the solution, suggesting its general applicability to any similar difference equation where the coefficients are constants and it's a linear form, providing a versatile and broadly applicable solution method.", 'The problem has a recursive structure, leading to first step analysis. (Relevance: 5)', 'The specific strategy is to condition on the outcome of the first round. (Relevance: 4)', 'The need to find something to condition on to simplify the problem. (Relevance: 3)', 'Deriving a recursive equation for solving the problem. (Relevance: 2)', "Importance of recognizing the problem's structure and thinking conditionally. (Relevance: 1)", 'The plan for today is to cover a famous important example of thinking conditionally and then start on random variables, setting the stage for the upcoming lectures.']}