A box contains 3 white and 5 black balls. Four balls are drawn at random

"with replacement" from the box (i.e. selected balls are returned to the
box and mixed, before drawing the next ball). Let X be the number of
black balls drawn. What is the distribution of X? What is the probability
that 3 or more black balls are drawn?

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To determine the distribution of X, we need to calculate the probability of getting each possible outcome. In this case, X can take values from 0 to 4 because there are at most 4 black balls in the box.

To calculate the probability of getting a certain number of black balls, we need to consider the probability of getting a black ball on each draw. Since the balls are drawn with replacement, the probability of drawing a black ball on each draw remains constant.

Let's break down the probability calculations for each outcome:

P(X = 0) = P(no black ball on any of the 4 draws)
To calculate this probability, we need to find the probability of drawing a white ball on each of the 4 draws. Since there are 3 white balls and 8 total balls in the box, the probability of getting a white ball on each draw is 3/8. Therefore, the probability of no black ball is (3/8)^4.

P(X = 1) = P(1 black ball and 3 white balls drawn)
To calculate this probability, we need to find the probability of getting one black ball and three white balls in any order. Since each draw is independent and the probability of getting a black ball is 5/8, the probability of getting exactly one black ball is (5/8)^1 * (3/8)^3. However, the black ball can appear in any of the four draws, so we multiply this probability by 4.

P(X = 2) = P(2 black balls and 2 white balls drawn)
Similarly, the probability of getting exactly two black balls is (5/8)^2 * (3/8)^2, and since there are six different ways this can happen (choosing 2 out of the 4 draws), we multiply this probability by 6.

P(X = 3) = P(3 black balls and 1 white ball drawn)
Similarly, the probability of getting exactly three black balls is (5/8)^3 * (3/8)^1, and since there are four different ways this can happen (choosing 3 out of the 4 draws), we multiply this probability by 4.

P(X = 4) = P(4 black balls drawn)
The probability of getting four black balls is (5/8)^4.

To find the probability that 3 or more black balls are drawn, we need to calculate the sum of the probabilities for X = 3 and X = 4:

P(X >= 3) = P(X = 3) + P(X = 4)

After calculating all these probabilities, we can determine the distribution of X and the probability of drawing 3 or more black balls.