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Expected value of binomial distribution | Probability and Statistics | Khan Academy

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{'title': 'Expected value of binomial distribution | Probability and Statistics | Khan Academy', 'heatmap': [{'end': 414.72, 'start': 260.206, 'weight': 0.86}], 'summary': 'Covers the expected value of a binomial distribution, explaining the general formula for the random variable x and the expected value being n times p or p times n, illustrated through the example of making basketball shots with a 40% shot percentage and taking 10 shots.', 'chapters': [{'end': 241.558, 'segs': [{'end': 69.426, 'src': 'embed', 'start': 39.163, 'weight': 4, 'content': [{'end': 46.848, 'text': 'And then we actually calculated the expected value for the particular binomial distributions that we studied,', 'start': 39.163, 'duration': 7.685}, {'end': 49.79, 'text': 'especially the one with the flipping of the coin.', 'start': 46.848, 'duration': 2.942}, {'end': 57.836, 'text': "In this video, we'll find a general formula for the expected value of a binomial distribution.", 'start': 50.15, 'duration': 7.686}, {'end': 69.426, 'text': 'So if we say that the random variable x The random variable x is equal to the number of successes with probability p after n trials.', 'start': 57.896, 'duration': 11.53}], 'summary': 'Calculated expected value for binomial distributions, focusing on coin flipping.', 'duration': 30.263, 'max_score': 39.163, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y39163.jpg'}, {'end': 153.985, 'src': 'embed', 'start': 112.299, 'weight': 0, 'content': [{'end': 115.82, 'text': 'you get that nice binomial distribution that looks a little bit like a bell curve.', 'start': 112.299, 'duration': 3.521}, {'end': 117.761, 'text': "And we'll study more about bell curves later.", 'start': 115.84, 'duration': 1.921}, {'end': 121.642, 'text': "And before I actually show it to you, I'll give you the answer.", 'start': 118.76, 'duration': 2.882}, {'end': 125.244, 'text': 'Because the answer, to some degree, is actually pretty intuitive.', 'start': 121.962, 'duration': 3.282}, {'end': 133.368, 'text': 'The expected value of this random variable is n times p.', 'start': 125.884, 'duration': 7.484}, {'end': 135.289, 'text': 'Or sometimes people will write p times n.', 'start': 133.368, 'duration': 1.921}, {'end': 137.41, 'text': 'Let me make that a little bit more tangible for you.', 'start': 135.289, 'duration': 2.121}, {'end': 153.985, 'text': "So if I said that x is equal to the number of baskets I make, or I'm talking about basketball, not basket weaving.", 'start': 137.99, 'duration': 15.995}], 'summary': 'Studying bell curves, expected value is n times p.', 'duration': 41.686, 'max_score': 112.299, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y112299.jpg'}, {'end': 222.223, 'src': 'embed', 'start': 172.869, 'weight': 1, 'content': [{'end': 192.788, 'text': "all I have to do is multiply the probability times, the number of baskets I'm taking, which would be equal to 4..", 'start': 172.869, 'duration': 19.919}, {'end': 199.071, 'text': "So I know I said, and you really shouldn't necessarily strictly view expected value as the number of shots you should expect to make,", 'start': 192.788, 'duration': 6.283}, {'end': 202.433, 'text': 'because sometimes probability distributions can be kind of weird.', 'start': 199.071, 'duration': 3.362}, {'end': 205.234, 'text': 'But in the binomial distribution, you can kind of view it that way.', 'start': 202.853, 'duration': 2.381}, {'end': 207.976, 'text': 'That this is the number of shots you would expect to make.', 'start': 205.574, 'duration': 2.402}, {'end': 211.077, 'text': 'Or you can kind of view it as the most likely outcome.', 'start': 208.356, 'duration': 2.721}, {'end': 218.321, 'text': "That if you have a 40% shot percentage, and you take 10 shots, the most likely outcome is that you'll make 4 shots.", 'start': 211.097, 'duration': 7.224}, {'end': 222.223, 'text': 'You still might make 6 shots or 3 shots, but this is going to be the most likely outcome.', 'start': 218.341, 'duration': 3.882}], 'summary': 'Using binomial distribution, 40% shot percentage and 10 shots would likely result in 4 made shots.', 'duration': 49.354, 'max_score': 172.869, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y172869.jpg'}], 'start': 0.632, 'title': 'Expected values of binomial distribution', 'summary': 'Discusses the expected value of a binomial distribution, with a general formula for the random variable x and the expected value being n times p or p times n. it also explains the concept of expected value using the example of making basketball shots, where with a 40% shot percentage and taking 10 shots, the most likely outcome is making 4 shots, calculated using the binomial distribution.', 'chapters': [{'end': 135.289, 'start': 0.632, 'title': 'Expected value of binomial distribution', 'summary': 'Discusses the expected value of a binomial distribution, with a general formula for the random variable x and the expected value being n times p or p times n.', 'duration': 134.657, 'highlights': ['The expected value of a binomial distribution is n times p or p times n.', 'Calculation of expected value for particular binomial distributions, especially the one with the flipping of the coin.', 'Explanation of the probability distribution for the random variable x with a nice binomial distribution that resembles a bell curve.']}, {'end': 241.558, 'start': 135.289, 'title': 'Expected value in basketball', 'summary': 'Explains the concept of expected value using the example of making basketball shots, where with a 40% shot percentage and taking 10 shots, the most likely outcome is making 4 shots, calculated using the binomial distribution.', 'duration': 106.269, 'highlights': ['The most likely outcome of making 4 shots after taking 10 shots with a 40% shot percentage is explained using the concept of expected value in the binomial distribution.', 'Expected value can be viewed as the number of shots one would expect to make, or as the most likely outcome in the binomial distribution, despite potential variability in actual outcomes.']}], 'duration': 240.926, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y632.jpg', 'highlights': ['The expected value of a binomial distribution is n times p or p times n.', 'The most likely outcome of making 4 shots after taking 10 shots with a 40% shot percentage is explained using the concept of expected value in the binomial distribution.', 'Explanation of the probability distribution for the random variable x with a nice binomial distribution that resembles a bell curve.', 'Expected value can be viewed as the number of shots one would expect to make, or as the most likely outcome in the binomial distribution, despite potential variability in actual outcomes.', 'Calculation of expected value for particular binomial distributions, especially the one with the flipping of the coin.']}, {'end': 1013.869, 'segs': [{'end': 414.72, 'src': 'heatmap', 'start': 260.206, 'weight': 0.86, 'content': [{'end': 266.292, 'text': "And I know it just gets a little complicated sometimes, but I'm just saying what's the probability in this basketball analogy?", 'start': 260.206, 'duration': 6.086}, {'end': 269.655, 'text': "What's the probability that I make?", 'start': 266.312, 'duration': 3.343}, {'end': 271.817, 'text': 'k could be 3 shots or something like that?', 'start': 269.655, 'duration': 2.162}, {'end': 272.858, 'text': "So that's what we're talking about.", 'start': 271.877, 'duration': 0.981}, {'end': 278.583, 'text': "And that, we learned, was if we're taking n shots, we're going to choose k of them.", 'start': 272.898, 'duration': 5.685}, {'end': 282.202, 'text': 'We did that several times in the last couple of videos.', 'start': 279.961, 'duration': 2.241}, {'end': 287.305, 'text': 'And then we multiply that times the probability of any one of those particular occurrences.', 'start': 282.823, 'duration': 4.482}, {'end': 296.009, 'text': "So if I'm making k shots, it'll be the probability of me making any one shot, which is p, to the k-th power.", 'start': 287.685, 'duration': 8.324}, {'end': 297.95, 'text': 'p times itself k times.', 'start': 296.109, 'duration': 1.841}, {'end': 299.831, 'text': "That's the probability of me making k shots.", 'start': 297.97, 'duration': 1.861}, {'end': 302.572, 'text': 'And then the rest of the shots I have to miss.', 'start': 300.531, 'duration': 2.041}, {'end': 305.274, 'text': 'So the probability of a miss is 1 minus p.', 'start': 302.913, 'duration': 2.361}, {'end': 310.942, 'text': "And then how many shots? If I've made k shots, the rest of the shots I have to miss.", 'start': 306.519, 'duration': 4.423}, {'end': 314.244, 'text': "So I'm going to miss n minus k shots.", 'start': 311.542, 'duration': 2.702}, {'end': 320.729, 'text': 'So in any binomial distribution, this is the probability that you get k successes.', 'start': 314.865, 'duration': 5.864}, {'end': 328.494, 'text': 'And we know that the expected value, the way you calculate an expected value of a random variable, is you just take the probability weighted sum.', 'start': 322.23, 'duration': 6.264}, {'end': 331.095, 'text': "And I don't want to confuse you too much.", 'start': 329.675, 'duration': 1.42}, {'end': 335.91, 'text': "And if your main takeaway from this video is just this, That's good enough.", 'start': 331.176, 'duration': 4.734}, {'end': 336.651, 'text': 'You should feel good.', 'start': 335.95, 'duration': 0.701}, {'end': 341.653, 'text': "Now it'll get a little technical, but it'll hopefully make you a little bit more comfortable with sigma and sum notation as well.", 'start': 337.091, 'duration': 4.562}, {'end': 345.155, 'text': "It'll make you a little bit more comfortable with binomial coefficients and things like that.", 'start': 341.713, 'duration': 3.442}, {'end': 350.077, 'text': 'But just going back, the expected value is the probability weighted sum of each of these.', 'start': 346.356, 'duration': 3.721}, {'end': 357.641, 'text': "So what you want to do is you want to take the probability that x is equal to k times k, and then add that up for each of the possible k's.", 'start': 350.097, 'duration': 7.544}, {'end': 367.525, 'text': "So how would I write that? So the expected value of x, the expected value of our random variable that's being described as binomial distribution.", 'start': 357.661, 'duration': 9.864}, {'end': 368.605, 'text': "It's equal to the sum.", 'start': 367.585, 'duration': 1.02}, {'end': 373.386, 'text': "And we're going to sum all the values that k can take on.", 'start': 370.905, 'duration': 2.481}, {'end': 376.127, 'text': 'So k can start at 0.', 'start': 373.766, 'duration': 2.361}, {'end': 377.967, 'text': "That's in the basketball version.", 'start': 376.127, 'duration': 1.84}, {'end': 378.907, 'text': 'I make no shots.', 'start': 378.007, 'duration': 0.9}, {'end': 382.228, 'text': 'All the way to n, which means I make n shots.', 'start': 379.447, 'duration': 2.781}, {'end': 386.469, 'text': 'And for each of them, you want to multiply k.', 'start': 383.208, 'duration': 3.261}, {'end': 387.329, 'text': 'So the outcome.', 'start': 386.469, 'duration': 0.86}, {'end': 390.11, 'text': 'So I made k shots times the probability that I make k shots.', 'start': 387.389, 'duration': 2.721}, {'end': 393.829, 'text': 'Well, what was the probability that I make k shots? That was this right here.', 'start': 390.13, 'duration': 3.699}, {'end': 404.874, 'text': "So it'd be k times n, choose k, times p to the k, times 1 minus p to the n minus k.", 'start': 394.529, 'duration': 10.345}, {'end': 409.917, 'text': "And now we're just going to do some algebra, some sigma algebra, I guess you could call it.", 'start': 404.874, 'duration': 5.043}, {'end': 414.72, 'text': "So the first simplification we can make is we're summing from k equals 0 to n.", 'start': 410.818, 'duration': 3.902}], 'summary': 'Probability of making k shots in n attempts explained with binomial distribution and expected value calculation.', 'duration': 154.514, 'max_score': 260.206, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y260206.jpg'}, {'end': 336.651, 'src': 'embed', 'start': 287.685, 'weight': 1, 'content': [{'end': 296.009, 'text': "So if I'm making k shots, it'll be the probability of me making any one shot, which is p, to the k-th power.", 'start': 287.685, 'duration': 8.324}, {'end': 297.95, 'text': 'p times itself k times.', 'start': 296.109, 'duration': 1.841}, {'end': 299.831, 'text': "That's the probability of me making k shots.", 'start': 297.97, 'duration': 1.861}, {'end': 302.572, 'text': 'And then the rest of the shots I have to miss.', 'start': 300.531, 'duration': 2.041}, {'end': 305.274, 'text': 'So the probability of a miss is 1 minus p.', 'start': 302.913, 'duration': 2.361}, {'end': 310.942, 'text': "And then how many shots? If I've made k shots, the rest of the shots I have to miss.", 'start': 306.519, 'duration': 4.423}, {'end': 314.244, 'text': "So I'm going to miss n minus k shots.", 'start': 311.542, 'duration': 2.702}, {'end': 320.729, 'text': 'So in any binomial distribution, this is the probability that you get k successes.', 'start': 314.865, 'duration': 5.864}, {'end': 328.494, 'text': 'And we know that the expected value, the way you calculate an expected value of a random variable, is you just take the probability weighted sum.', 'start': 322.23, 'duration': 6.264}, {'end': 331.095, 'text': "And I don't want to confuse you too much.", 'start': 329.675, 'duration': 1.42}, {'end': 335.91, 'text': "And if your main takeaway from this video is just this, That's good enough.", 'start': 331.176, 'duration': 4.734}, {'end': 336.651, 'text': 'You should feel good.', 'start': 335.95, 'duration': 0.701}], 'summary': 'Probability of making k shots in binomial distribution explained.', 'duration': 48.966, 'max_score': 287.685, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y287685.jpg'}, {'end': 1011.227, 'src': 'embed', 'start': 983.611, 'weight': 0, 'content': [{'end': 988.01, 'text': 'And so that is going to be equal to 1.', 'start': 983.611, 'duration': 4.399}, {'end': 993.674, 'text': "And so we're left with the expected value of our random variable.", 'start': 988.01, 'duration': 5.664}, {'end': 1000.819, 'text': 'x is equal to n times p, where n is the number of trials and p is the probability of success on each trial.', 'start': 993.674, 'duration': 7.145}, {'end': 1003.922, 'text': 'And this is true only for binomial distributions.', 'start': 1000.839, 'duration': 3.083}, {'end': 1005.843, 'text': "This isn't true for any random variable x.", 'start': 1003.942, 'duration': 1.901}, {'end': 1011.227, 'text': 'Only true for a random variable x whose probability distribution is the binomial distribution.', 'start': 1005.843, 'duration': 5.384}], 'summary': 'Expected value of random variable x equals n times p for binomial distributions.', 'duration': 27.616, 'max_score': 983.611, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y983611.jpg'}], 'start': 242.218, 'title': 'Binomial distribution in basketball', 'summary': 'Discusses the probability of making k shots out of n in a binomial distribution, where k and n represent specific numbers of shots and trials, and p represents the probability of making a shot.', 'chapters': [{'end': 314.244, 'start': 242.218, 'title': 'Binomial distribution in basketball', 'summary': 'Discusses the probability of making k shots out of n in a binomial distribution, where k and n represent specific numbers of shots and trials, and p represents the probability of making a shot.', 'duration': 72.026, 'highlights': ['The probability of making k shots out of n in a binomial distribution is calculated by choosing k shots out of n, multiplying it by the probability of making a shot to the power of k, and multiplying it by the probability of missing a shot to the power of (n-k).', 'The probability of making a shot, denoted as p, is raised to the power of k to represent the probability of making k shots, while the probability of missing a shot, denoted as 1 - p, is raised to the power of (n - k) to represent the probability of missing the remaining shots.']}, {'end': 1013.869, 'start': 314.865, 'title': 'Expected value of binomial distribution', 'summary': 'Explains the expected value of a random variable in a binomial distribution, demonstrating how it equals n times p, where n is the number of trials and p is the probability of success on each trial, and it is true only for binomial distributions.', 'duration': 699.004, 'highlights': ['The expected value of a random variable in a binomial distribution is equal to n times p, where n is the number of trials and p is the probability of success on each trial.', 'The expected value calculation involves summing the probability weighted sum for each possible value of k, which ranges from 0 to n in the context of the binomial distribution.', 'The simplification of the expected value expression leads to the result np, indicating that the expected value of a random variable in a binomial distribution is np.']}], 'duration': 771.651, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/SqcxYnNlI3Y/pics/SqcxYnNlI3Y242218.jpg', 'highlights': ['The expected value of a random variable in a binomial distribution is equal to n times p, where n is the number of trials and p is the probability of success on each trial.', 'The probability of making k shots out of n in a binomial distribution is calculated by choosing k shots out of n, multiplying it by the probability of making a shot to the power of k, and multiplying it by the probability of missing a shot to the power of (n-k).', 'The expected value calculation involves summing the probability weighted sum for each possible value of k, which ranges from 0 to n in the context of the binomial distribution.']}], 'highlights': ['The expected value of a binomial distribution is n times p or p times n.', 'The most likely outcome of making 4 shots after taking 10 shots with a 40% shot percentage is explained using the concept of expected value in the binomial distribution.', 'The probability of making k shots out of n in a binomial distribution is calculated by choosing k shots out of n, multiplying it by the probability of making a shot to the power of k, and multiplying it by the probability of missing a shot to the power of (n-k).', 'Explanation of the probability distribution for the random variable x with a nice binomial distribution that resembles a bell curve.', 'Expected value can be viewed as the number of shots one would expect to make, or as the most likely outcome in the binomial distribution, despite potential variability in actual outcomes.']}