Question

A lady claimed that she is able to tell whether the tea was added first or the milk was added first to a cup. To test this idea, a statistician proposed to give her ten cups of tea, each made in random order (tea first or milk first) without telling her which is which. Assume each cup is independent. Let X equal to the number of cups that the lady identified correctly.

(a) Suppose that she is just randomly guessing, with a 50-50 percent chance. Find P(X = 7) .

(b) Suppose again that she is randomly guessing, with a 50-50 percent chance. This experiment will be stopped early if she cannot correctly identify at least one cup among the first three cups. What is the probability that this experiment continues beyond three cups?

(c) Suppose she is indeed able to tell 9% of the time. Find the probability she correctly identifies at least 7 cups.

Answer 1

a) The probability of getting any trial correct is here 0.5 as there is random guessing. Therefore the probability here is computed using the binomial probability function as:

therefore 0.1171875 is the required probability here.

b) Probability that the experiment continues beyond three cups

= 1 - Probability that all the three guesses are incorrect

= 1 - 0.5³

= 1 - 0.125

= 0.875

Therefore 0.875 is the required probability here.

c) As she is able to tell 9% of the time, and rest 91% of the time is random guessing, therefore the probability of getting it correct is computed here as:
= 0.09 + 0.91*0.5

= 0.545

The probability that she correctly identifies at least 7 cups is computed here as:

Therefore 0.2556 is the required probability here.