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 obviously the best strategy. At every turn he chooses the suit that is most likely to be next, thus maximizing the expected correct guesses on that turn. This is guaranteed to maximize the total expected correct guesses.

I am not going to tackle the first bonus question yet, as it seems the most difficult. Let's try the original question first.