Suppose that there is an urn that contains black balls and white balls . You randomly draw balls with replacement . Answer the following questions with explanation.
(1) Find the probability that you draw a black ball for the first time at the -th draw .
(2) Suppose that you have drawn a black ball for the first time at the -th draw. Find the probability that you draw one or more black balls in the remaining draws.
Let be a random variable the value of which is 1 if the -th ball is black, and 0 otherwise . If necessary, you can use the equalities , and .
To find the probability of drawing a black ball for the first time at the -th draw, we need to calculate the probability of drawing white balls in the first draws and a black ball on the -th draw.
Let be the probability of drawing a black ball in any given draw, and be the probability of drawing a white ball.
The probability of the first draws being white is , and the probability of the -th draw being black is . Hence, the probability of this event is:
Given that the first black ball is drawn at the -th draw, we want to find the probability of drawing at least one more black ball in the remaining draws.
The probability of not drawing a black ball in any of the remaining draws is . Therefore, the probability of drawing at least one black ball in the remaining draws is:
The random variable indicates whether the -th ball is black. Since the draws are independent, the expected value of is simply the probability of drawing a black ball in a single draw: