Question

1. There is a deck of 12 cards numbered 1 through 12 . You draw a card from the deck, look at it, and then put it back in the deck. You draw a total of two cards in this way. Let's define the following events: - A: You get an even-numbered card and an odd-numbered card regardless of their order. - B: The first card you draw is even numbered. - C: The sum of the cards is even. - D: The sum of the cards is 18. (a) Are events A and B independent? (b) Are events A and C independent? (c) Compute P(D). (d) Compute P(D∣C).

Answer 1

Answer and Step-by-step explanation:

(a) To determine if events A and B are independent, we need to check if the probability of event A occurring is affected by the occurrence of event B, and vice versa.

Event A: You get an even-numbered card and an odd-numbered card regardless of their order.

Event B: The first card you draw is even numbered.

Since event A involves both an even-numbered card and an odd-numbered card, event B is a subset of event A. In other words, if event B occurs (the first card is even), it guarantees that event A will also occur. Therefore, events A and B are not independent.

(b) Similarly, to determine if events A and C are independent, we need to check if the probability of event A occurring is affected by the occurrence of event C, and vice versa.

Event C: The sum of the cards is even.

Event A involves the occurrence of both an even-numbered card and an odd-numbered card. The sum of an even number and an odd number is always odd. Therefore, if event C occurs (the sum is even), event A cannot occur. Thus, events A and C are not independent.

(c) To compute P(D), we need to find the probability of event D occurring.

Event D: The sum of the cards is 18.

To calculate P(D), we need to determine the number of favorable outcomes and the total number of possible outcomes.

In this case, there is only one favorable outcome: drawing the cards numbered 9 and 9. Since there are 12 cards in the deck, the total number of possible outcomes is 12 * 12 = 144 (assuming all cards are equally likely to be drawn).

Therefore, P(D) = favorable outcomes / total outcomes = 1/144.

(d) To compute P(D∣C), we need to find the conditional probability of event D occurring given that event C has already occurred.

Event D: The sum of the cards is 18.

Event C: The sum of the cards is even.

Since event C has already occurred (the sum is even), we know that the sum of the two cards is either 14 or 16. For event D to occur, the sum must be exactly 18.

There is only one favorable outcome: drawing the cards numbered 9 and 9. As mentioned earlier, the total number of possible outcomes is 144.

Therefore, P(D∣C) = favorable outcomes / total outcomes = 1/144.