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.

(b) Evaluate the expectation E

To evaluate the expectation E, we need to find the expected value of ⇠.

The number of new balls among the selected ones, ⇠, can take values from 0 to 4 inclusive. We need to calculate the probability of each possible outcome and then multiply it by the value of ⇠ for that outcome. Finally, we sum up all the products to get the expected value.

Let's break down the calculation step by step:

Step 1: Calculate the probability for each possible outcome:

- If 0 new balls are selected: This means all 4 balls chosen are previously used. The probability of this is given by: (4 used balls / 9 total balls) * (3 used balls / 8 remaining balls) * (2 used balls / 7 remaining balls) * (1 used ball / 6 remaining balls).

- If 1 new ball is selected: This means exactly 1 of the chosen balls is new and the other 3 are previously used. The probability of this is given by: (5 new ball / 9 total balls) * (4 used balls / 8 remaining balls) * (3 used balls / 7 remaining balls) * (2 used balls / 6 remaining balls).

- If 2 new balls are selected: This means exactly 2 of the chosen balls are new and the other 2 are previously used. The probability of this is given by: (5 new balls / 9 total balls) * (4 new balls / 8 remaining balls) * (3 used balls / 7 remaining balls) * (2 used balls / 6 remaining balls).

- If 3 new balls are selected: This means exactly 3 of the chosen balls are new and the other 1 is previously used. The probability of this is given by: (5 new balls / 9 total balls) * (4 new balls / 8 remaining balls) * (3 new balls / 7 remaining balls) * (2 used balls / 6 remaining balls).

- If 4 new balls are selected: This means all 4 balls chosen are new. The probability of this is given by: (5 new balls / 9 total balls) * (4 new balls / 8 remaining balls) * (3 new balls / 7 remaining balls) * (2 new balls / 6 remaining balls).

Step 2: Multiply each probability by the respective value of ⇠:

- For 0 new balls, multiply the probability by 0.
- For 1 new ball, multiply the probability by 1.
- For 2 new balls, multiply the probability by 2.
- For 3 new balls, multiply the probability by 3.
- For 4 new balls, multiply the probability by 4.

Step 3: Sum up all the products from step 2 to get the expected value E:

E = (Probability of 0 new balls * 0) + (Probability of 1 new ball * 1) + (Probability of 2 new balls * 2) + (Probability of 3 new balls * 3) + (Probability of 4 new balls * 4).

By performing these calculations, you will find the expected value E for the number of new balls among the selected ones.