Re: D01-38
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30 May 2016, 13:43
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?
So, we have :
Sample of tea A : the cups are A1, A2, and A3
Sample of tea B : the cups are B1, B2, and B3
Sample of tea C : the cups are C1, C2, and C3
So the contestant is offered : 4 random cups from the set {A1, A2, A3, B1, B2, B3, C1, C2, C3}
We want to calculate the probability to not choose the 3 samples with 4 cups ; for instance A1A2A3B2, or B13BC1C2, or C2A1A2A3, or A1A2B2B3, etc.
->So the contestant could test ONLY 2 of the 3 samples
Proba = \(\frac{(Number.of.ways.to.choose.4.cups.from.2.samples)}{(Number.total.of.ways.to.choose.4.cups.from.3.samples)}\)
1/ Number of ways to choose 4 cups from 2 samples :
Stage 1 : Choose 2 samples from the 3 = \(\frac{3*2}{2}\) = 3
We obtain : A, A, A, B, B, B for example
Stage 2 : Choose 4 from the 6 cups selected = \(\frac{6*5*4*3}{4*3*2}\) = 15
Counting Principle : 3*15 ways
2/ Number of ways to choose 4 cups from 3 samples = from 9 cups :
\(\frac{9*8*7*6}{4*3*2}\) = 9*7*2
So, proba = \(\frac{3*15}{9*7*2}\) = \(\frac{5}{14}\)