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 hearts.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To calculate the expected number of pairs of adjacent cards that are both hearts, we can start by calculating the probability that any two adjacent cards are both hearts.

Since there are a total of 52 cards in the deck, there are 13 hearts. Therefore, the probability that any one card is a heart is 13/52 = 1/4.

To calculate the probability that two adjacent cards are both hearts, we can multiply the probability that the first card is a heart by the probability that the second card is also a heart. Since the two events are independent, we can multiply the probabilities:

(1/4) * (1/4) = 1/16

Therefore, the probability that two adjacent cards are both hearts is 1/16.

Since there are 52 cards in total, there are 52 pairs of adjacent cards. Therefore, the expected number of pairs of adjacent cards that are both hearts is:

52 * (1/16) = 3.25

Therefore, the expected number of pairs of adjacent cards that are both hearts is 3.25.