An unbiased die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a six and a five.
Find (a) E[X],
(b) E[X|Y = 1]
In this blog post, we will discuss the expectations of the number of rolls necessary to obtain a six (X) and a five (Y) in a sequence of successive rolls of an unbiased die. We will calculate (a) E[X], (b) E[X|Y=1], and (c) E[X|Y=5].
Calculating E[X]:
The expected value of X is the average number of rolls necessary to obtain a six. Let's assume that the probability of obtaining a six in one roll is 1/6. Therefore, the probability of not obtaining a six in one roll is 5/6. To calculate E[X], we can use the formula:
E[X] = 1/P(X=1) + 2/P(X=2) + 3/P(X=3) + ...
Where P(X=n) is the probability of obtaining a six in n rolls.
P(X=1) = 1/6, P(X=2) = (5/6) * (1/6), P(X=3) = (5/6)^2 * (1/6), ...
Therefore, the expected value of X is:
E[X] = 1/1/6 + 2/(5/6) * (1/6) + 3/(5/6)^2 * (1/6) + ...
E[X] = 6.
Calculating E[X|Y=1]:
The expected value of X given Y=1 is the average number of rolls necessary to obtain a six, given that a five was obtained in the first roll. To calculate E[X|Y=1], we use the formula:
E[X|Y=1] = 1/P(X=1|Y=1) + 2/P(X=2|Y=1) + 3/P(X=3|Y=1) + ...
Where P(X=n|Y=1) is the probability of obtaining a six in n rolls given that a five was obtained in the first roll.
P(X=1|Y=1) = 0, P(X=2|Y=1) = 1/6, P(X=3|Y=1) = (5/6) * (1/6), ...
Therefore, the expected value of X given Y=1 is:
E[X|Y=1] = 1/0 + 2/1/6 + 3/(5/6) * (1/6) + ...
E[X|Y=1] = 7.