Question

a standard deck of cards contains 52 cards. these 52 cards are arranged in a circle, at random. find the expected number of pairs of adjacent cards that are both black.

Answer 1

The expected number of pairs of adjacent cards that are both black in a random arrangement of a standard deck of cards is 13.

Step-by-step explanation:

To find the expected number of pairs of adjacent cards that are both black in a deck of 52 cards arranged in a random circle, we need to calculate the probability of each adjacent pair being black.

First, let's calculate the probability of the first card being black. There are 26 black cards in a deck of 52, so the probability is 26/52 = 1/2.

Next, let's calculate the probability of the second card being black, given that the first card is black. If the first card is black, there are now 25 black cards left in the deck of 51 remaining cards, so the probability is 25/51.

Since the two cards are adjacent, we can multiply the probabilities together to get the probability of the pair being black: (1/2) * (25/51) = 25/102.

Since there are 52 cards in the deck, there are 52 possible pairs. Therefore, the expected number of pairs of adjacent cards that are both black is (25/102) * 52 = 13.