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probability tea

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by jzebra10 » Tue Nov 29, 2011 12:09 am
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!
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by Anurag@Gurome » Tue Nov 29, 2011 12:49 am
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.
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by rijul007 » Tue Nov 29, 2011 12:55 am
Total no of ways of tasting 4 cups of tea = 9P4 = 6*7*8*9

No of ways that all three samples are selected =>
Selecting two of a kind = 3*3C2 = 9
Other two cups different types = 3*3 = 9
Arranging them = 4!

Total no of ways = 9*9*4!

Probability that a contestant tastes all samples = P1 = 9*9*4!/6*7*8*9 = 9/14

probability that a contestant does not taste all of the samples = 1-P1 = 1- 9/14 = 5/14

IMO: B
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by shankar.ashwin » Tue Nov 29, 2011 12:58 am
You could also do it directly,

Since there are only 3 cups of each type, we can be sure he tastes 2 varieties of tea (Since he tastes a total of 4).

We have 3 samples and we taste only 2 - 3C2 = 3 ways of choosing which 2 to taste

And once we select the 2 samples, we have 6 cups of the 2 samples and we pick 4 - 6C4 = 15

Total ways = 9C4 = 126

So, we have (3*15)/126 = 5/14 (3C2*6C4/9C4)
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by HSPA » Tue Nov 29, 2011 12:58 am
Base: 4 out of 9 has to be chosen = 9C4

Does not taste all flavours = 1- tastes all flavours

Conditions to taste all flavours = 3*( 3c1*3c1*3c2) = 3^4

result = 1- 3^4/ 9c4 = 1- 9/14 = 5/14
First take: 640 (50M, 27V) - RC needs 300% improvement
Second take: coming soon..
Regards,
HSPA.
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