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

This pigeonhole principle is among mathematics' simplest but then most useful principles, and it can help us here.
A simplified form states if (N+1) pigeons inhabit N holes, then each hole must contain at least two pigeons. If 5 pigeons occupied 4 holes, there must be at least one hole from at least 2 pigeons.

There are 20 balls, 50 per cent of which are red and half of which are blue.

Part a: least three balls of the same color:

The pigeonholes are now the colors x/2 must equal three, and the smallest positive integer that will satisfy this equation is 5.

Part b: least three blue balls;

Because the first 10 selections could all consist of red balls, the woman must select a minimum of 13 balls to ensure that at least three of them are blue.

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The correct question is-

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?