2. Pierre-Simon Laplace(1749-1827) wanted to answer the question: what is the probability that the sun will rise tomorrow? Thus, he conducted the following imaginary experiment.

Setting. There are N + 1 boxes, whose numbers are from 0 to N. In the box with the number k (k = 0, 1 . . . , N), there are k red balls and N k blue balls. You cannot see the number on each box, and every box looks exactly the same. That is, you cannot distinguish the boxes.

Experiment. First, you randomly choose one box. Then, you randomly draw one ball in that box with replacement n times. Surprisingly, all n balls are red. Now, you are going to draw a ball one more time from the same box. We want to calculate the probability that this next ball is also red. Let's call this probability p, and call this ball (n + 1)-th ball. Assume N > 1 and n > 0. (Note: There must be N, n, or k in your answers.)

(a) [1pt] Let's define several events:

Ak = {Choosing box k} (k = 0, 1, . . . , n), B = {All n balls are red}, C = {The next (n + 1)-th ball is red}.

What is P(Ak) for k = (0, 1, . . . N)?

(b) [1pt] What are P(B|Ak) and P(BC|Ak)? (Hint: Recall the meaning of the conditional probability. You can think that you have already chosen the box k.)

(c) [2pts] What are P(B) and P(B C)? (Hint: What would your partition be?)

(d) [1pt] Calculate p using the result of (c).