Question

You and I are playing a game. You start each round with 8 cards where four are red, and four are black. YOu then select 6 cards, without letting me see the color, and place them face down. Finally, I guess the color of the majority of the cards. I can guess "Red" if I think 4 more cards are red, "Black" if I believe 4 or more are black, or "Even" if I believe there are 3 of each color. If I guess right, I get one point, if I guess wrong, you get one point. Your strategy, which you belive is the best strategy is to randomly select the six cards from your 8 cards.

A. what is the probability that you will select the same number of each color

B. What is the probability that you will select all red cards.

C. What is the probability of you selecting a majority of red cards.

Answer 1

In total there are 8 cards, fir black and four red. I'm selecting six cards from this deck. If X is the number of red cards then X follows a hypergeometric distribution. The pmf of X is given by:

a) probability that you will select the same number of each color i.e. P(X=3), then using above formula we have:

b) probability that you will select all red cards i.e.

c) probability of you selecting a majority of red cards i.e. you picked 4 red cards. Hence using previous problem we know the probability is