There are 9 tennis balls in a box, of which only 5 have not previously

been used. Four of the balls are randomly chosen from the box. Let � be the
number of new balls among the selected ones.
(a) Find the probability distribution of �:
(b) Evaluate the expectation E�:
(c) Evaluate the variance V ar �

To find the probability distribution of �, we need to consider the possible values it can take on and calculate the probability corresponding to each value.

(a) Probability distribution of �:
Let's consider the possible values of �: 0, 1, 2, 3, 4.

When � = 0, it means all four chosen balls have been previously used. The probability of this occurring can be determined by selecting 4 balls from the 5 used balls divided by selecting 4 balls from the 9 total balls in the box:

P(� = 0) = (5 Choose 4) / (9 Choose 4)

When � = 1, it means exactly one out of the four chosen balls is new. The probability of this occurring can be determined by selecting 1 new ball and 3 used balls from the available options:

P(� = 1) = (5 Choose 1) * (4 Choose 3) / (9 Choose 4)

Similarly, the probabilities for � = 2, 3, and 4 can be calculated in the same way, taking into account the number of new balls and used balls selected:

P(� = 2) = (5 Choose 2) * (4 Choose 2) / (9 Choose 4)
P(� = 3) = (5 Choose 3) * (4 Choose 1) / (9 Choose 4)
P(� = 4) = (5 Choose 4) * (4 Choose 0) / (9 Choose 4)

(b) Evaluation of the expectation E�:
The expectation E� can be evaluated by multiplying each possible value of � by its corresponding probability and then summing them up:

E� = 0 * P(� = 0) + 1 * P(� = 1) + 2 * P(� = 2) + 3 * P(� = 3) + 4 * P(� = 4)

(c) Evaluation of the variance Var�:
The variance Var� can be evaluated using the formula:

Var� = E(�^2) - (E�)^2

where E(�^2) represents the expectation of the square of �. We can calculate E(�^2) using the same approach as in part (b), by multiplying each possible value of �^2 by its corresponding probability and then summing them up.

Finally, we can substitute the values obtained for E� and E(�^2) into the variance formula to calculate Var�.