Question

There are six blue balls and four red balls in the pocket. Take out a ball at random, check the color and put it back in the pocket.
1. If you take the ball out until the red one comes out, what is the probability that the ball will be drawn exactly five times and the experiment is over?
2. What is the average and variance of the number of times X is taken if the ball is pulled out until the red ball comes out?
3. Repeat the procedure ten times to remove the ball from the pocket. Find the mean and variance of the number of blue balls taken Y.

Answer 1

There are six blue balls and four red balls in the pocket.

(1) Here in the initital four pickups we draw one blue ball and in fifth pickup we draw red ball. Here we are taking out balls with replacement as we are putting it back in the pocket.

so,

P(The ball will be drawn exactly five times) = (6/10) * (6/10) * (6/10) * (6/10) * (4/10) = 0.05184

(2) Here Probability of taking red ball in an attempt = p = 4/10

so expected number of number of times X is taken untile the red ball comes out = 1/(4/10) = 2.5

Variance of number of times X is taken until the red ball comes out = (1-p)/p² = (1 - 4/10)/ (4/10)² = (6/10)/ (16/100) = 15/4

(3) Here Expected number of blue balls are taken would be one less than expected number of times would be less than expected value of X.

Expected number of blue balls taken Y = E[Y] = 2.5 -1 = 1.5

Variance of number of blue balls take Y = VaR[Y] = VAR[X] = 15/4

so now we take the procedure ten times to remove the ball from the pocket

so mean number of blue balls taken = 1.5

variance of number of blue balls taken Y = (15/4) * (1/10) = 15/40 = 3/8