Question

Suppose a deck of 52 cards is shuffled and the top two cards are dealt.

1. How many ordered pairs of cards could possibly results as outcomes?
2. Assuming each of theses pairs has the same chances, calculate the chance that the first card is an ace.
3. Under the same assumption, calculate the chance that the second card is an ace.
4. Under the same assumption, calculate the chance that both cards are aces.
5. Under the same assumption, calculate the chance of at least one ace among the two cards.

Answer 1

The number of possible ordered pairs of cards from a shuffled deck is 2652. The chance that the first card is an ace is 1/13. The chance that the second card is an ace is 1/12.

Step-by-step explanation:

1. To find the number of possible ordered pairs of cards, we can use the concept of permutations. The first card can be any of the 52 cards, and the second card can be any of the remaining 51 cards. Therefore, the total number of ordered pairs is 52 × 51 = 2652.

2. Since there are 4 aces in the deck, the chance that the first card is an ace is 4/52 = 1/13.

3. Assuming the first card was not an ace (remaining 48 non-ace cards in the deck), the chance that the second card is an ace is 4/48 = 1/12.

4. To calculate the chance that both cards are aces, we multiply the individual probabilities. Therefore, the chance that both cards are aces is (1/13) × (1/12) = 1/156.

5. The chance of at least one ace among the two cards can be calculated as 1 minus the chance that neither card is an ace. The probability that the first card is not an ace is 48/52 = 12/13. Similarly, the probability that the second card is not an ace, given that the first card was not an ace, is 47/51. Therefore, the chance of at least one ace is 1 - (12/13) × (47/51) = 119/221.