4. a bowl contains 10 red balls and 10 blue balls. a woman selects balls at random without looking at them.
a) how many balls must she select to be sure of having at least three balls of the same color?
b) how many balls must she select to be sure of having at least three blue balls?

a) 5 balls

b) 13 balls

Step-by-step explanation:

a) by the pigeon holes theorem where the pigeon is the ball and the holes here are the colors. For her to have at least 3 balls of the same colors, we can consider the worst case where she selected 2 balls of each color, that makes it 4 balls. Then by the 5th ball it should match either of the colors, making it 3 balls of same color.

b) Similarly, in the worst case let's say she selects all 10 red balls, then the next 3 balls must be blue. So she needs to select a total of 13 balls to make sure that she has at least 3 blue balls.