The Gardner Sisters

Puzzle:

Gretchen and Henry invited the four Gardner sisters over to their house for an afternoon tea. Henry went to the cabinet to take out some plates (they have both blue and green plates in the cabinet), and the first two plates he took out were blue. "What are the odds?" asked Martina Gardner, the youngest.

Henry thought for a moment, and then replied, "Knowing how many plates of each color I have, the probability that I would pull out two blue ones is exactly 1/2!" Martina then asked if Henry could feed two dozen people if he used all of his plates. "Not quite," he replied.

She then told him how many plates he had. What number did she say?

Solution:

Henry has 21 plates, and 15 of them are blue.

This makes the probability of drawing two plates (15/21) * (14/20), which equals 1/2.

Martina had to ask if he could feed two dozen people because there are other (larger) numbers that work. For example, if he had 85 blue plates out of 120 total, the probability that he would pull out two blue ones would have been (85/120) * (84/119), or 1/2. Other numbers work, as well, but all are greater than 120.