Step 1: Understand the problem. The problem is related to finding the mean of the probability distribution for the number of red balls drawn when 3 balls are drawn from an urn containing 3 black and 5 red balls. The number of red balls that can be drawn ranges from 0 to 3, and we need to calculate the expected value (mean) of this distribution.
Step 2: Calculating probabilities. The number of possible outcomes when drawing 3 balls from the urn is: Next, we calculate the probability of drawing 0, 1, 2, and 3 red balls. - For 0 red balls, all 3 balls must be black. The number of ways to choose 3 black balls is: So the probability is: - For 1 red ball, we need to choose 1 red ball and 2 black balls. The number of ways to do this is: So the probability is: - For 2 red balls, we need to choose 2 red balls and 1 black ball. The number of ways to do this is: So the probability is: - For 3 red balls, all 3 balls must be red. The number of ways to choose 3 red balls is: So the probability is: Step 3: Calculating the expected value. The expected value (mean) of the number of red balls drawn is the sum of each outcome multiplied by its probability: Simplifying: Thus, the mean of the probability distribution of the number of red balls drawn is .
