1. The problem asks for the distribution that models the number on a card selected at random from 20 cards numbered 1 to 20. 2. Since each card is equally likely to be selected, the distribution is a uniform distribution over the integers 1 to 20. 3. This means the probability of selecting any particular card number $k$ is $P(X=k) = \frac{1}{20}$ for $k=1,2,\ldots,20$. 4. The reason is that all outcomes are equally likely and discrete, so the uniform discrete distribution applies. 5. Next, we have a discrete random variable $X$ with values $4,8,16$ and probabilities $0.3,0.5,0.2$ respectively. 6. Two independent values of $X$ are generated: call them $X_1$ and $X_2$. 7. We want to find $P(X_1 \leq X_2)$. 8. Since $X_1$ and $X_2$ are independent, $P(X_1 = a, X_2 = b) = P(X_1 = a)P(X_2 = b)$. 9. List all pairs $(a,b)$ where $a,b \in \{4,8,16\}$ and $a \leq b$: - $(4,4), (4,8), (4,16), (8,8), (8,16), (16,16)$ 10. Calculate each probability: - $P(4,4) = 0.3 \times 0.3 = 0.09$ - $P(4,8) = 0.3 \times 0.5 = 0.15$ - $P(4,16) = 0.3 \times 0.2 = 0.06$ - $P(8,8) = 0.5 \times 0.5 = 0.25$ - $P(8,16) = 0.5 \times 0.2 = 0.10$ - $P(16,16) = 0.2 \times 0.2 = 0.04$ 11. Sum these probabilities: $$0.09 + 0.15 + 0.06 + 0.25 + 0.10 + 0.04 = 0.69$$ 12. Therefore, the probability that the first value is less than or equal to the second value is $0.69$.