You are allowed to take the test up to N = 10 times. After the score of a test comes out, you may either submit it or take the test again, but you may not submit the score of an old test. Your scores are independent from test to test. Denoted by X i your score of the i th test. Your best score of all N tests is B. Assume that B may occur in any test at an equal probability.

(a) If you submit your score of the first test, what is the probability that it is the best score B?

(b) Suppose that you record your score X 1 of the first test, take more tests one by one, and submit your score whenever it is better than X 1. What is the probability that the submitted score B 1 is the best score B?

(c) Suppose that you record your best score, denoted by S N/2, in the first N/2 tests, take more tests one by one, and submit your score whenever it is better than S N/2. What is the probability that the submitted score B N/2 is the best score B?

(d) Suppose that you record your best score, denoted by S k, in the first k tests, take more tests one by one, and submit your score whenever it is better than S k. What is the probability that the submitted score B k is the best score B?

(e) In (d), as k increases, how does the probability P{Bk = B} vary?

(f) What is the best k to use?