Question: Once upon a time, a famous statistician ordered tea to a lady. The lady claimed that she could tell whether milk had been added to the cup before or after the tea. The statistician decided to run some experiments to test her claim.
(a) The lady is given 6 cups of tea, where it is known in advance that 3 will be milk- first and 3 will be tea-first, in a completely random order. The lady gets to taste each and then guess which 3 were milk-first. Assume for this part that she has no ability whatsoever to distinguish milk-first from tea-first cups of tea. Find the probability that at least 2 of her 3 guesses are correct.
(b) Now the lady is given one cup of tea, with probability 1/2 of it being milk-first. She needs to say whether she thinks it was milk-first. Let p1 be the lady's probability of being correct given that it was milk-first, and p2 be her probability of being correct given that it was tea-first. She claims that the cup was milk-first. Find the posterior odds that the cup is milk-first, given this information.