probability tea

jzebra10 wrote:I don't understand the explanation given in the book. Can someone try to explain this better?

At a blind taste competition a contestant is offered 3 cups of each of the 3 samples of tea in a random arrangement of 9 marked cups. If each contestant tastes 4 different cups of tea, what is the probability that a contestant does not taste all of the samples?

a) 1/12
b) 5/14
c) 4/9
d) 1/2
e) 2/3

thanks!

Probability that a contestant does not taste all of the samples = 1 - probability that a contestant tastes all 3 samples.

This means that first we should find the probability that a contestant should taste 2 cups of one sample and 1 cup from each of 2 other samples.

Total no. of ways to choose 4 cups out of 9 = 9C4
No. of ways to choose the sample which provides with 2 cups = 3C1
No. of ways to choose these 2 cups from the sample chosen = 3C2
No. of ways to choose 1 cup out of 3 from the 2nd sample = 3C1
No. of ways to choose 1 cup out of 3 from the 3rd sample = 3C1

So, probability that a contestant tastes all 3 samples = (3C1 * 3C2 * 3C1 * 3C1)/9C4 = (3 * 3 * 3 * 3)/(18 * 7) = 9/14

Therefore, required probability = 1 - 9/14 = [spoiler]5/14[/spoiler]

The correct answer is B.