Question

Once upon a time, a famous statistician offered 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 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 pi 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.

Answer 1

Combining combinatorics and probability, we can find the chance of the lady guessing at least 2 out of 3 cups correctly by chance alone. In the Bayesian inference problem, we update our beliefs about whether milk was added first using the lady's accuracy rates and her claim.

Step-by-step explanation:

The student is presented with problems in probability, specifically involving the concepts of random guessing and the use of Bayesian inference to update the odds of an event.

If the lady guesses at random without any ability to distinguish, her guessing is similar to random picking without replacement, reminiscent of a hypergeometric distribution. However, she only needs to guess correctly and is not ordering her guesses. So, we have to calculate the probability of her guessing at least 2 out of 3 cups correctly purely by chance. There are 20 different ways she can pick 3 out of the 6 cups (combinations of 6 taken 3 at a time, written as C(6, 3) or 6 Choose 3). The lady can guess correctly in the following scenarios: guessing all 3 correctly, guessing 2 correctly and 1 incorrectly, or guessing 1 correctly and 2 incorrectly. We need to find the probabilities for these scenarios and add them up.

For Bayesian inference, we calculate the posterior odds that the cup is milk-first given the lady's claim. Using pi and p2, we assume a prior probability of 1/2 for milk-first or tea-first. We adjust these odds by the likelihood of the lady making the correct call, which requires plugging in the values for pi and p2 accordingly. The posterior odds are then calculated as the product of the prior odds and the likelihood ratio.