In a sequence of n + k independent Bernoulli trials with success probability p in each trial, let X be the number of successes in the first n trials and Y be the total number of successes out of the n + k trials. Establish that for 0 ≤ x ≤ y, 0 ≤ x ≤ n, and 0 ≤ y ≤ n + k, the following expressions are true:

How do you interpret the last result intuitively?

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(i) P(X = X, Y = y) = (") (yx) pqy. -X (ii) P(X = y | X = x) = ( x ) p-xqx (iii) P(X x|Y=y): n k (A)(5) (***) y