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One day you overhear how two fellow students discuss a show in which a performer spins a coin many mes, resulting in Tails 30 mes in a row. You hear them discussing: Yueh Han: "That outcome would be extremely unlikely with fair coins. They must be using trick coins (maybe double-tailed coins), or the experiment must have been rigged somehow (maybe with magnets)." Akma: "It's true that the string TT...T of length 30 is very unlikely; the chance is (12) 30 ~ 9 × 10-10 with fair coins. But any other specific string of H and T with length 30 has exactly the same probability! The reason the outcome seems extremely unlikely is that the number of possible outcomes grows exponen ally as the number of spins grows, so any outcome would seem extremely unlikely. You could just as well have made the same argument even without looking at the results of their experiment, which means you really don't have evidence against the coins being fair." Help Yueh Han and Akma resolve their debate. 1. Discuss the conversation of Yueh Han and Akma. With whom do you agree and why? 2. Suppose there are only two models: either the coins are all fair (and spun fairly), or double-tailed coins are being used in which case the probability of Tails is 1. Let p be the prior probability that the coins are fair. Find the posterior probability that the coins are fair, given that they landed Tails in 30 out of 30 trials. 3. For which values of the prior, p, is the posterior probability that the coins are fair greater than 0.5? (What does the prior need to be to make the fair-coin model more probable according to the posterior?)

One day you overhear how two fellow students discuss a show in which a performer spins a coin many mes, resulting in Tails 30 mes in a row. You hear them discussing: Yueh Han: "That outcome would be extremely unlikely with fair coins. They must be using trick coins (maybe double-tailed coins), or the experiment must have been rigged somehow (maybe with magnets)." Akma: "It's true that the string TT...T of length 30 is very unlikely; the chance is (12) 30 ~ 9 × 10-10 with fair coins. But any other specific string of H and T with length 30 has exactly the same probability! The reason the outcome seems extremely unlikely is that the number of possible outcomes grows exponen ally as the number of spins grows, so any outcome would seem extremely unlikely. You could just as well have made the same argument even without looking at the results of their experiment, which means you really don't have evidence against the coins being fair." Help Yueh Han and Akma resolve their debate. 1. Discuss the conversation of Yueh Han and Akma. With whom do you agree and why? 2. Suppose there are only two models: either the coins are all fair (and spun fairly), or double-tailed coins are being used in which case the probability of Tails is 1. Let p be the prior probability that the coins are fair. Find the posterior probability that the coins are fair, given that they landed Tails in 30 out of 30 trials. 3. For which values of the prior, p, is the posterior probability that the coins are fair greater than 0.5? (What does the prior need to be to make the fair-coin model more probable according to the posterior?)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.3: Conditional Probability; Independent Events; Bayes' Theorem
Problem 39E: The following problem submitted by Daniel Hahn of Blairstown, Iowa, appeared in the Ask Marilyn...
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Solve 1, 2, and 3. Show reasoning and explian calculations and the topics. 

One day you overhear how two fellow students discuss a show in which a performer spins a coin many mes, resulting in Tails 30 mes in a row. You hear them discussing: Yueh Han: "That outcome would be extremely unlikely with fair coins. They must be using trick coins (maybe double-tailed coins), or the experiment must have been rigged somehow (maybe with magnets)." Akma: "It's true that the string TT...T of length 30 is very unlikely; the chance is (1) 30 9 × 10-10 with fair coins. But any other specific string of H and T with length 30 has exactly the same probability! The reason the outcome seems extremely unlikely is that the number of possible outcomes grows exponen ally as the number of spins grows, so any outcome would seem extremely unlikely. You could just as well have made the same argument even without looking at the results of their experiment, which means you really don't have evidence against the coins being fair." Help Yueh Han and Akma resolve their debate. 1. Discuss the conversation of Yueh Han and Akma. With whom do you agree and why? 2. Suppose there are only two models: either the coins are all fair (and spun fairly), or double-tailed coins are being used in which case the probability of Tails is 1. Let p be the prior probability that the coins are fair. Find the posterior probability that the coins are fair, given that they landed Tails in 30 out of 30 trials. 3. For which values of the prior, p, is the posterior probability that the coins are fair greater than 0.5? (What does the prior need to be to make the fair-coin model more probable according to the posterior?)
Transcribed Image Text:One day you overhear how two fellow students discuss a show in which a performer spins a coin many mes, resulting in Tails 30 mes in a row. You hear them discussing: Yueh Han: "That outcome would be extremely unlikely with fair coins. They must be using trick coins (maybe double-tailed coins), or the experiment must have been rigged somehow (maybe with magnets)." Akma: "It's true that the string TT...T of length 30 is very unlikely; the chance is (1) 30 9 × 10-10 with fair coins. But any other specific string of H and T with length 30 has exactly the same probability! The reason the outcome seems extremely unlikely is that the number of possible outcomes grows exponen ally as the number of spins grows, so any outcome would seem extremely unlikely. You could just as well have made the same argument even without looking at the results of their experiment, which means you really don't have evidence against the coins being fair." Help Yueh Han and Akma resolve their debate. 1. Discuss the conversation of Yueh Han and Akma. With whom do you agree and why? 2. Suppose there are only two models: either the coins are all fair (and spun fairly), or double-tailed coins are being used in which case the probability of Tails is 1. Let p be the prior probability that the coins are fair. Find the posterior probability that the coins are fair, given that they landed Tails in 30 out of 30 trials. 3. For which values of the prior, p, is the posterior probability that the coins are fair greater than 0.5? (What does the prior need to be to make the fair-coin model more probable according to the posterior?)
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