### Brain Teasers

# The Gardner Sisters II

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

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?

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?

### Hint

He has more than 6 plates, so he could feed everyone, but less than 24. There is only one number in the middle of these that would satisfy the "exactly half" requirement.### Answer

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.

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## Comments

i like this teaser very much!

Why couldn't it be 1 green and 3 blue plates?

Because they have enough for themselves and for the 4 sisters.

oh ok

I wrote a BASIC program and finally came up with the answer.

I figured out first that they could have 4 plates of which 3 are blue so the probability would be 3/4 * 2/3 = 1/2. But then I figured that since they said "not quite" when asked about feeding two dozen people, that you were probably looking for a larger number.

You can get all the possibilities by graphing or just making a table of xy points from the equations:

y/x * (y-1)/(x-1) = 1/2

x < 24

where y=blue plates and x=total plates

and see which points from x=2 (at least 2 plates) to x=23 will give a whole positive number for y.

x=4 and x=21 are the only two for x < 24.

You can get all the possibilities by graphing or just making a table of xy points from the equations:

y/x * (y-1)/(x-1) = 1/2

x < 24

where y=blue plates and x=total plates

and see which points from x=2 (at least 2 plates) to x=23 will give a whole positive number for y.

x=4 and x=21 are the only two for x < 24.

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