title

The Binomial Distribution and Test, Clearly Explained!!!

description

The binomial distribution and the related statistical test look really complicated, but a actually quite simple. Here I walk you through both, one step at a time, so that they are easily understood and applied.
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detail

{'title': 'The Binomial Distribution and Test, Clearly Explained!!!', 'heatmap': [{'end': 361.024, 'start': 328.089, 'weight': 0.73}, {'end': 436.856, 'start': 424.554, 'weight': 0.737}, {'end': 725.292, 'start': 661.399, 'weight': 0.749}, {'end': 876.876, 'start': 829.468, 'weight': 0.782}], 'summary': 'Clearly explains binomial distribution and test using a survey to determine preference between orange and grape fanta, emphasizing sample size and random chance. it explores modeling preference with binomial distribution, discussing the probability of orange fanta preference among people and calculating probabilities, concluding that both flavors are equally loved.', 'chapters': [{'end': 301.75, 'segs': [{'end': 72.905, 'src': 'embed', 'start': 36.211, 'weight': 3, 'content': [{'end': 39.972, 'text': 'A coin usually has heads and at least one tail.', 'start': 36.211, 'duration': 3.761}, {'end': 48.193, 'text': 'For example, you can use the binomial distribution to find out the probability of getting six heads in six tosses.', 'start': 41.072, 'duration': 7.121}, {'end': 59.535, 'text': 'But who really cares about flipping coins? What folks really want to know is whether or not people like orange Fanta more than grape Fanta.', 'start': 49.813, 'duration': 9.722}, {'end': 72.905, 'text': 'Which flavor reigns supreme? Or are they both equally loved? To answer this question, we can ask a bunch of people which flavor they prefer.', 'start': 60.902, 'duration': 12.003}], 'summary': 'Using binomial distribution, determine the probability of getting six heads in six coin tosses, while also exploring the preference between orange and grape fanta.', 'duration': 36.694, 'max_score': 36.211, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E836211.jpg'}, {'end': 175.282, 'src': 'embed', 'start': 115.523, 'weight': 0, 'content': [{'end': 121.226, 'text': 'To get to the bottom of this mystery, we need to get a sense of what to expect if there is no preference.', 'start': 115.523, 'duration': 5.703}, {'end': 126.989, 'text': 'Then we determine if our survey results fit those expectations.', 'start': 123.207, 'duration': 3.782}, {'end': 132.091, 'text': 'If not, we can reject the idea that both Fantas are loved equally.', 'start': 127.729, 'duration': 4.362}, {'end': 138.555, 'text': 'The binomial distribution will tell us what to expect if there is no preference.', 'start': 134.032, 'duration': 4.523}, {'end': 142.837, 'text': 'To say the same thing using statistics lingo.', 'start': 140.175, 'duration': 2.662}, {'end': 153.902, 'text': 'We will use the binomial distribution, aka this nasty looking thing, to model what to expect when there is no preference.', 'start': 143.851, 'duration': 10.051}, {'end': 158.047, 'text': "Then we'll see how well this model fits the data.", 'start': 155.464, 'duration': 2.583}, {'end': 165.315, 'text': 'If the model is a poor fit, we will reject the idea that both flavors are loved equally.', 'start': 159.669, 'duration': 5.646}, {'end': 175.282, 'text': "So let's start with a super simple example and assume that I asked three people if they liked orange Fanta more than grape Fanta.", 'start': 167.219, 'duration': 8.063}], 'summary': 'Using binomial distribution to test preference for fanta flavors.', 'duration': 59.759, 'max_score': 115.523, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8115523.jpg'}, {'end': 236.98, 'src': 'embed', 'start': 205.992, 'weight': 4, 'content': [{'end': 213.916, 'text': 'We can then calculate the probability of the first two people randomly choosing orange and the third person randomly choosing grape.', 'start': 205.992, 'duration': 7.924}, {'end': 222.172, 'text': 'Assuming that there is no real preference, the probability of the first person preferring orange Fanta is 0.5.', 'start': 215.649, 'duration': 6.523}, {'end': 236.98, 'text': 'And the probability of the first two people preferring orange Fanta is 0.5 times 0.5, which equals 0.25.', 'start': 222.172, 'duration': 14.808}], 'summary': 'Probability of first two people choosing orange fanta is 0.25.', 'duration': 30.988, 'max_score': 205.992, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8205992.jpg'}], 'start': 0.436, 'title': 'Binomial distribution & test', 'summary': 'Explains binomial distribution and test, using survey results to determine preference between orange and grape fanta, emphasizing sample size and random chance. it also explores modeling preference with binomial distribution, providing a simple example and calculating probabilities.', 'chapters': [{'end': 113.515, 'start': 0.436, 'title': 'Binomial distribution & test', 'summary': 'Explains the binomial distribution and binomial test, using the example of determining the preference between orange and grape fanta based on survey results, highlighting the importance of sample size and random chance.', 'duration': 113.079, 'highlights': ['The binomial distribution is used to find the probability of getting a certain outcome in a fixed number of independent trials, such as the probability of getting six heads in six coin tosses.', 'The chapter emphasizes the relevance of the binomial test in determining preferences, as demonstrated by the example of survey results for orange and grape Fanta, showcasing the importance of sample size and random chance in drawing conclusions.']}, {'end': 301.75, 'start': 115.523, 'title': 'Preference modeling with binomial distribution', 'summary': 'Explores using the binomial distribution to model the probability of people preferring one fanta flavor over the other, demonstrating a simple example and calculating the probabilities.', 'duration': 186.227, 'highlights': ['The binomial distribution is used to model the probability of people preferring one Fanta flavor over the other, with a simple example demonstrating the calculation of probabilities.', 'The chapter explains how to calculate the probability of people randomly choosing different Fanta flavors assuming there is no real preference, emphasizing that all combinations are equally likely.', 'The concept of using the binomial distribution to model expectations when there is no preference is introduced, with a demonstration of calculating probabilities based on equal preference assumptions.']}], 'duration': 301.314, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8436.jpg', 'highlights': ['The chapter emphasizes the relevance of the binomial test in determining preferences, as demonstrated by the example of survey results for orange and grape Fanta, showcasing the importance of sample size and random chance in drawing conclusions.', 'The binomial distribution is used to model the probability of people preferring one Fanta flavor over the other, with a simple example demonstrating the calculation of probabilities.', 'The concept of using the binomial distribution to model expectations when there is no preference is introduced, with a demonstration of calculating probabilities based on equal preference assumptions.', 'The binomial distribution is used to find the probability of getting a certain outcome in a fixed number of independent trials, such as the probability of getting six heads in six coin tosses.', 'The chapter explains how to calculate the probability of people randomly choosing different Fanta flavors assuming there is no real preference, emphasizing that all combinations are equally likely.']}, {'end': 542.082, 'segs': [{'end': 361.024, 'src': 'heatmap', 'start': 303.531, 'weight': 0, 'content': [{'end': 312.016, 'text': 'And this means that the probability that any two out of three people prefer orange Fanta is the sum of the three possible orders.', 'start': 303.531, 'duration': 8.485}, {'end': 316.319, 'text': 'So we just add the three probabilities together.', 'start': 313.737, 'duration': 2.582}, {'end': 328.089, 'text': 'and the probability that any 2 out of 3 people would randomly say they prefer orange Fanta is 0.375.', 'start': 318.086, 'duration': 10.003}, {'end': 332.53, 'text': 'Alternatively, we could have done the math using this nasty-looking formula.', 'start': 328.089, 'duration': 4.441}, {'end': 337.852, 'text': 'x is the number of people who preferred orange Fanta.', 'start': 334.811, 'duration': 3.041}, {'end': 342.633, 'text': 'In this case, x equals 2.', 'start': 338.512, 'duration': 4.121}, {'end': 344.994, 'text': 'n is the total number of people we asked.', 'start': 342.633, 'duration': 2.361}, {'end': 349.602, 'text': 'In this case, n equals 3.', 'start': 345.634, 'duration': 3.968}, {'end': 354.623, 'text': 'N minus X the total number of people we asked minus.', 'start': 349.602, 'duration': 5.021}, {'end': 361.024, 'text': 'the number of people who preferred orange Fanta equals the number of people who said they prefer grape Fanta.', 'start': 354.623, 'duration': 6.401}], 'summary': 'The probability that any 2 out of 3 people prefer orange fanta is 0.375.', 'duration': 28.999, 'max_score': 303.531, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8303531.jpg'}, {'end': 470.324, 'src': 'heatmap', 'start': 424.554, 'weight': 1, 'content': [{'end': 436.856, 'text': 'And if we plug in n equals 3 and x equals 2, and then just do the math, we get 3.', 'start': 424.554, 'duration': 12.302}, {'end': 445.243, 'text': 'Three ways that two out of three people could prefer orange Fanta, just like when we did it by hand.', 'start': 436.856, 'duration': 8.387}, {'end': 449.466, 'text': 'So this fancy thing is really no big deal.', 'start': 446.884, 'duration': 2.582}, {'end': 460.135, 'text': 'The next part of the formula, p to the x, corresponds to the probability that orange Fanta was chosen two of the three times.', 'start': 451.528, 'duration': 8.607}, {'end': 470.324, 'text': 'In other words, p to the x just consolidates 0.5 times 0.5 into 0.5 squared.', 'start': 461.776, 'duration': 8.548}], 'summary': 'Probability calculation shows 3 ways 2 out of 3 prefer orange fanta.', 'duration': 102.739, 'max_score': 424.554, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8424554.jpg'}], 'start': 303.531, 'title': 'Fanta preference probabilities', 'summary': 'Discusses the probability of orange fanta preference among three people, resulting in a probability of 0.375 and also explores the probability of preferences for orange and grape fanta, showcasing 3 ways for 2 out of 3 people to prefer orange fanta and a 0.5 probability of someone preferring grape fanta.', 'chapters': [{'end': 404.021, 'start': 303.531, 'title': 'Probability of orange fanta preference', 'summary': 'Explains the probability of two out of three people preferring orange fanta, resulting in a probability of 0.375, and presents a formula to calculate the probability based on the number of people and the probability of picking orange fanta.', 'duration': 100.49, 'highlights': ['The probability that any 2 out of 3 people would randomly say they prefer orange Fanta is 0.375, calculated by adding the three probabilities together.', 'The chapter presents a formula to calculate the probability of X, the number of people who say they prefer orange Fanta, given N, the number of people we asked, and P, the probability of picking orange Fanta.']}, {'end': 542.082, 'start': 404.867, 'title': 'Probability of fanta preferences', 'summary': 'Explains the probability of preferences for orange and grape fanta among three people, showing that there are 3 ways for 2 out of 3 people to prefer orange fanta and the probability of someone preferring grape fanta is 0.5.', 'duration': 137.215, 'highlights': ['There are 3 ways for 2 out of 3 people to prefer orange Fanta.', 'The probability of someone preferring grape Fanta is 0.5.', 'The formula for the probability of preferences for orange and grape Fanta consolidates 0.5 times 0.5 into 0.5 squared.']}], 'duration': 238.551, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8303531.jpg', 'highlights': ['The probability that any 2 out of 3 people would randomly say they prefer orange Fanta is 0.375, calculated by adding the three probabilities together.', 'The formula for the probability of preferences for orange and grape Fanta consolidates 0.5 times 0.5 into 0.5 squared.', 'There are 3 ways for 2 out of 3 people to prefer orange Fanta.', 'The chapter presents a formula to calculate the probability of X, the number of people who say they prefer orange Fanta, given N, the number of people we asked, and P, the probability of picking orange Fanta.', 'The probability of someone preferring grape Fanta is 0.5.']}, {'end': 748.329, 'segs': [{'end': 629.041, 'src': 'embed', 'start': 542.082, 'weight': 0, 'content': [{'end': 545.363, 'text': 'the probability someone would randomly pick orange Fanta.', 'start': 542.082, 'duration': 3.281}, {'end': 559.33, 'text': 'and we get the same probability that two out of three people would randomly prefer orange Fanta that we got when we did everything by hand, 0.375.', 'start': 546.823, 'duration': 12.507}, {'end': 560.59, 'text': 'In other words,', 'start': 559.33, 'duration': 1.26}, {'end': 568.535, 'text': 'the binomial distribution tells us that the probability that two of three people will prefer orange Fanta due to random chance is 0.375..', 'start': 560.59, 'duration': 7.945}, {'end': 584.257, 'text': 'Bam! Calculating the probability of 3 of 3 people saying they prefer orange Fanta by hand is pretty easy, since there is only one combination.', 'start': 568.535, 'duration': 15.722}, {'end': 592.002, 'text': 'But we can just as easily use the fancy formula by plugging in x equals 3.', 'start': 585.778, 'duration': 6.224}, {'end': 593.543, 'text': 'And then we just do the math.', 'start': 592.002, 'duration': 1.541}, {'end': 600.508, 'text': 'This term equals 1, since we are dividing 3 factorial by 3 factorial.', 'start': 595.064, 'duration': 5.444}, {'end': 609.575, 'text': 'And this term is also equal 1, because anything raised to the 0 power equals 1.', 'start': 602.113, 'duration': 7.462}, {'end': 611.356, 'text': 'And then we just keep doing the math.', 'start': 609.575, 'duration': 1.781}, {'end': 625.32, 'text': 'And this means that the probability of 3 of 3 people randomly preferring orange Fanta is 0.125,,', 'start': 615.577, 'duration': 9.743}, {'end': 629.041, 'text': 'which is exactly what we got when we did the calculations by hand.', 'start': 625.32, 'duration': 3.721}], 'summary': 'The probability of three out of three people preferring orange fanta is 0.125, as calculated using the binomial distribution.', 'duration': 86.959, 'max_score': 542.082, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8542082.jpg'}, {'end': 725.292, 'src': 'heatmap', 'start': 650.091, 'weight': 1, 'content': [{'end': 661.399, 'text': 'Now we plug in x equals 4, the number of people that preferred orange Fanta, n equals 7, the number of people we asked and p equals 0.5,', 'start': 650.091, 'duration': 11.308}, {'end': 664.761, 'text': 'the probability someone would randomly pick orange Fanta.', 'start': 661.399, 'duration': 3.362}, {'end': 667.943, 'text': 'And then just do the math.', 'start': 666.422, 'duration': 1.521}, {'end': 680.36, 'text': 'and we get 0.273, the probability that four of seven people would randomly prefer orange Fanta.', 'start': 672.475, 'duration': 7.885}, {'end': 691.346, 'text': "Double bam! When you use a binomial distribution to calculate a p-value, it's called a binomial test.", 'start': 682.121, 'duration': 9.225}, {'end': 698.17, 'text': "So what's the p-value for four out of seven people preferring orange Fanta?", 'start': 692.807, 'duration': 5.363}, {'end': 712.665, 'text': 'The p-value is the probability of the observed data, plus the probabilities of all other possibilities that are equally likely or rarer.', 'start': 699.858, 'duration': 12.807}, {'end': 717.227, 'text': 'This means we need to calculate these probabilities.', 'start': 714.506, 'duration': 2.721}, {'end': 725.292, 'text': 'These are the observed results of our poll, and these are rarer possibilities.', 'start': 719.068, 'duration': 6.224}], 'summary': 'Using binomial distribution, the probability of 4 out of 7 people preferring orange fanta is 0.273.', 'duration': 67.136, 'max_score': 650.091, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8650091.jpg'}], 'start': 542.082, 'title': 'Binomial probability for fanta preference', 'summary': 'Explains binomial probability for orange fanta preference among people, with probabilities of 0.375 and 0.125 for two out of three and three out of three people respectively, and a 0.273 probability for four out of seven people preferring orange fanta.', 'chapters': [{'end': 629.041, 'start': 542.082, 'title': 'Calculating binomial probability', 'summary': 'Explains the calculation of binomial probability for the preference of orange fanta among three people, stating that the probability of two out of three people preferring orange fanta due to random chance is 0.375, and the probability of three out of three people randomly preferring orange fanta is 0.125.', 'duration': 86.959, 'highlights': ['The binomial distribution tells us that the probability of two out of three people preferring orange Fanta due to random chance is 0.375.', 'The probability of three out of three people randomly preferring orange Fanta is 0.125.', 'Explanation of using the binomial formula to calculate the probability of three out of three people preferring orange Fanta.']}, {'end': 748.329, 'start': 630.99, 'title': 'Binomial test for fanta preference', 'summary': 'Explores the use of binomial distribution to calculate the probability and p-value for people preferring orange fanta, and discusses the concept of a binomial test, concluding that there is a 0.273 probability that four out of seven people would randomly prefer orange fanta.', 'duration': 117.339, 'highlights': ['The probability that four of seven people would randomly prefer orange Fanta is 0.273, calculated using the binomial distribution.', 'The p-value for four out of seven people preferring orange Fanta is determined by considering the observed data and all other possibilities that are equally likely or rarer.', 'The concept of a binomial test is introduced, which involves calculating the probabilities of observed results and rarer possibilities, such as 4 versus 3 and 3 versus 4.', 'Explanation of a binomial test and the calculation of p-value for Fanta preference based on observed and rarer possibilities.']}], 'duration': 206.247, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8542082.jpg', 'highlights': ['The probability of three out of three people randomly preferring orange Fanta is 0.125.', 'The probability that four of seven people would randomly prefer orange Fanta is 0.273, calculated using the binomial distribution.', 'The binomial distribution tells us that the probability of two out of three people preferring orange Fanta due to random chance is 0.375.', 'The concept of a binomial test is introduced, which involves calculating the probabilities of observed results and rarer possibilities, such as 4 versus 3 and 3 versus 4.', 'The p-value for four out of seven people preferring orange Fanta is determined by considering the observed data and all other possibilities that are equally likely or rarer.', 'Explanation of using the binomial formula to calculate the probability of three out of three people preferring orange Fanta.', 'Explanation of a binomial test and the calculation of p-value for Fanta preference based on observed and rarer possibilities.']}, {'end': 936.836, 'segs': [{'end': 876.876, 'src': 'heatmap', 'start': 820.486, 'weight': 0, 'content': [{'end': 827.59, 'text': 'Adding the probabilities together gives us 0.5, the probability that orange Fanta is not preferred.', 'start': 820.486, 'duration': 7.104}, {'end': 842.701, 'text': 'The sum of the probabilities of all combinations of events that have an equal probability or are rarer equals 0.5 plus 0.5 which equals 1.', 'start': 829.468, 'duration': 13.233}, {'end': 850.629, 'text': 'Which means the p-value for 4 out of 7 people saying they prefer orange Fanta is 1.', 'start': 842.701, 'duration': 7.928}, {'end': 857.431, 'text': 'which means that the model the binomial distribution with p equals 0.5, i.e..', 'start': 850.629, 'duration': 6.802}, {'end': 863.592, 'text': 'orange Fanta and grape Fanta are both equally loved is a good fit for the observed data.', 'start': 857.431, 'duration': 6.161}, {'end': 876.876, 'text': 'Thus, we conclude that, given the sample size, 7, we cannot rule out the possibility that both orange Fanta and grape Fanta are equally loved.', 'start': 865.653, 'duration': 11.223}], 'summary': 'Probability of orange fanta not preferred is 0.5; p-value for 4 out of 7 people preferring orange fanta is 1, indicating a good fit for binomial distribution with p=0.5.', 'duration': 30.143, 'max_score': 820.486, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8820486.jpg'}, {'end': 936.836, 'src': 'embed', 'start': 884.649, 'weight': 1, 'content': [{'end': 889.693, 'text': "Triple bam! One last thing before we're done.", 'start': 884.649, 'duration': 5.044}, {'end': 898.32, 'text': 'The binomial distribution only works when the probability that someone likes Orange Fanta does not change,', 'start': 891.254, 'duration': 7.066}, {'end': 901.242, 'text': 'if someone else already said they liked Orange Fanta.', 'start': 898.32, 'duration': 2.922}, {'end': 909.688, 'text': 'In other words, if we ask a bunch of people if they like Orange Fanta and they all say yes,', 'start': 902.785, 'duration': 6.903}, {'end': 915.111, 'text': 'then that should not affect the probability that the next person also likes Orange Fanta.', 'start': 909.688, 'duration': 5.423}, {'end': 920.834, 'text': "Hooray! We've made it to the end of another exciting StatQuest.", 'start': 916.792, 'duration': 4.042}, {'end': 925.156, 'text': 'If you like this StatQuest and want to see more of them, please subscribe.', 'start': 921.554, 'duration': 3.602}, {'end': 933.169, 'text': 'And if you want to support StatQuest, well, please click the like button below and consider buying one or two of my original songs.', 'start': 925.716, 'duration': 7.453}, {'end': 936.836, 'text': 'Alright, until next time, quest on!.', 'start': 933.81, 'duration': 3.026}], 'summary': "Binomial distribution works if liking orange fanta probability doesn't change with others' responses.", 'duration': 52.187, 'max_score': 884.649, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8884649.jpg'}], 'start': 750.72, 'title': 'Two-sided p-value and binomial distribution', 'summary': 'Explains the calculation of two-sided p-value for preference of fanta flavors, concluding that both orange fanta and grape fanta are equally loved, with a p-value of 1, and encourages support for statquest.', 'chapters': [{'end': 936.836, 'start': 750.72, 'title': 'Two-sided p-value and binomial distribution', 'summary': 'Explains the calculation of two-sided p-value for preference of fanta flavors, with a probability of 0.5, concluding that both orange fanta and grape fanta are equally loved based on the observed data, with a p-value of 1. it also highlights the condition for the binomial distribution to work and encourages the audience to subscribe and support statquest.', 'duration': 186.116, 'highlights': ['The probability that 4 out of 7 people prefer orange Fanta is 0.273, and after calculation, the p-value is 1, indicating that both orange Fanta and grape Fanta are equally loved based on the observed data.', 'The binomial distribution only works when the probability that someone likes Orange Fanta does not change, if someone else already said they liked Orange Fanta.', 'Encourages the audience to subscribe and support StatQuest by clicking the like button and considering buying original songs.']}], 'duration': 186.116, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/J8jNoF-K8E8/pics/J8jNoF-K8E8750720.jpg', 'highlights': ['The probability that 4 out of 7 people prefer orange Fanta is 0.273, with a p-value of 1.', 'The binomial distribution only works when the probability of liking Orange Fanta does not change.', 'Encourages the audience to subscribe and support StatQuest.']}], 'highlights': ['The binomial distribution is used to model the probability of people preferring one Fanta flavor over the other, with a simple example demonstrating the calculation of probabilities.', 'The chapter emphasizes the relevance of the binomial test in determining preferences, as demonstrated by the example of survey results for orange and grape Fanta, showcasing the importance of sample size and random chance in drawing conclusions.', 'The concept of using the binomial distribution to model expectations when there is no preference is introduced, with a demonstration of calculating probabilities based on equal preference assumptions.', 'The probability that any 2 out of 3 people would randomly say they prefer orange Fanta is 0.375, calculated by adding the three probabilities together.', 'The probability of someone preferring grape Fanta is 0.5.', 'The probability that four of seven people would randomly prefer orange Fanta is 0.273, calculated using the binomial distribution.', 'The binomial distribution tells us that the probability of two out of three people preferring orange Fanta due to random chance is 0.375.', 'The concept of a binomial test is introduced, which involves calculating the probabilities of observed results and rarer possibilities, such as 4 versus 3 and 3 versus 4.', 'The probability that 4 out of 7 people prefer orange Fanta is 0.273, with a p-value of 1.', 'The binomial distribution only works when the probability of liking Orange Fanta does not change.']}