Let's break down the problem in steps:

(a) Probability Distribution Table for $X$ (your profit)

• You choose 4 numbers from 1 to 14 without replacement.
• The operator also chooses 4 numbers from 1 to 14 without replacement.

Possible outcomes for profit $X$:

• Match exactly 2 numbers: You win $10.
• Match exactly 3 numbers: You win $100.
• Match all 4 numbers: You win $1000.
• Match less than 2 numbers: You lose $75.

We need to calculate the probability for each of these outcomes and create a probability distribution table. To compute probabilities, we'll use combinations.

Let’s define the total number of ways to select 4 numbers from 14: $\text{Total possibilities} = \binom{14}{4} = \frac{14!}{4!(14-4)!} = 1001$

Now, for each scenario:

1. Match 4 numbers: You win $1000. This happens if all your 4 numbers match the operator’s numbers. The probability is: $\binom{4}{4} \binom{10}{0} / \binom{14}{4} = 1/1001$

2. Match 3 numbers: You win $100. You match 3 of your 4 numbers, and 1 of the operator's remaining numbers is not yours: $\binom{4}{3} \binom{10}{1} / \binom{14}{4} = \frac{4 \times 10}{1001} = 40/1001$

3. Match 2 numbers: You win $10. You match 2 of your 4 numbers, and the other 2 operator's numbers are not yours: $\binom{4}{2} \binom{10}{2} / \binom{14}{4} = \frac{6 \times 45}{1001} = 270/1001$

4. Match less than 2 numbers: You lose $75. This is the remaining probability: $1 - \left(\frac{1 + 40 + 270}{1001}\right) = \frac{690}{1001}$

Now, let’s fill in the probability table: