A player draws the cards from the a 52-card deck one by one, without putting them back in the deck.

Every time before drawing a card he guesses the suit of the card he will draw.

He decides to always guess the suit that occurs most frequently in the remaining deck (if there are
several such suits, he chooses any one of them).

Prove that he will guess the right suit at least 13
times.

This is a trick question. He

**won't** guess the right suit at least 13 times.

Spoiler:

Here comes the solution

The expected number of successes, E[X]=kp, where

k is the number of guesses

p os the probability of a correct guess

is equal to 13. This is an expected value, a mean. For any given run of trials, the actual number of correct guesses is more or less. The standard deviation is given by

(kp(1-p))^(1/2) = 3.12

68% of the time the number of correct guesses will be between about 10 and 16.

95% of the time the number of correct guesses will be between about 4 and 22.