Understanding the Probabilities of Drawing a Pair in a Deck of Playing Cards

Two cards are randomly selected from a deck of 52 playing cards.

(a) What is the probability they constitute a pair (that is, that they are of the same

denomination)?

(b) What is the conditional probability they constitute a pair given that they are

of different suits?

In this article, we'll explore the probabilities of drawing a pair of cards from a deck of 52 playing cards. A pair is defined as two cards of the same denomination. We'll also calculate the conditional probability of drawing a pair given that they are of different suits.

Assumptions:

The deck of playing cards contains 52 cards.

Two cards are selected randomly and without replacement.

(a) To find the probability of drawing a pair, we need to find the number of favorable outcomes (a pair) and divide it by the total number of possible outcomes (52 choose 2, or C(52,2)).

There are 13 denominations in a deck of playing cards, and each denomination has 4 cards. Therefore, there are 4 * 13 = 52 cards in the deck.

The number of favorable outcomes (a pair) can be calculated as follows:

13 * C(4,2) = 13 * 6 = 78

The total number of possible outcomes is C(52,2) = 26 * 51 / 2 = 1326

Therefore, the probability of drawing a pair is 78 / 1326 = 0.0588.

(b) To find the conditional probability of drawing a pair given that they are of different suits, we need to find the number of favorable outcomes (a pair of different suits) and divide it by the number of possible outcomes (two cards of different suits).

There are C(52,2) = 1326 ways to choose two cards from the deck, and C(13,2) = 78 ways to choose two different denominations. However, since we are only considering different suits, we need to divide the number of ways to choose two different denominations by 4 (the number of suits).

The number of favorable outcomes (a pair of different suits) can be calculated as follows:

78 / 4 = 19.5

The number of possible outcomes (two cards of different suits) is C(52,2) - C(4,2) = 1326 - 78 = 1248

Therefore, the conditional probability of drawing a pair given that they are of different suits is 19.5 / 1248 = 0.0157.

We have calculated the probability of drawing a pair from a deck of 52 playing cards and the conditional probability of drawing a pair given that they are of different suits. This analysis can be extended to other card games where the goal is to draw a pair of cards. Understanding the probabilities of drawing a pair is crucial in making decisions and strategies in these games.