At a blind taste competition a contestant is offered 3 cups : GMAT Problem Solving (PS)
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# At a blind taste competition a contestant is offered 3 cups

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At a blind taste competition a contestant is offered 3 cups [#permalink]

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14 Nov 2009, 06:34
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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. $$\frac{1}{12}$$
B. $$\frac{5}{14}$$
C. $$\frac{4}{9}$$
D. $$\frac{1}{2}$$
E. $$\frac{2}{3}$$
[Reveal] Spoiler: OA

Last edited by Bunuel on 06 Mar 2013, 01:30, edited 2 times in total.
Edited the question and added the OA
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14 Nov 2009, 07:06
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Economist wrote:
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?

# $$\frac{1}{12}$$
# $$\frac{5}{14}$$
# $$\frac{4}{9}$$
# $$\frac{1}{2}$$
# $$\frac{2}{3}$$

And the good one again. +1 to Economist.

"The probability that a contestant does not taste all of the samples" means that contestant tastes only 2 samples of tea (one sample is not possible as contestant tastes 4 cups>3 of each kind).

$$\frac{C^2_3*C^4_6}{C^4_9}=\frac{5}{14}$$.

$$C^2_3$$ - # of ways to choose which 2 samples will be tasted;
$$C^4_6$$ - # of ways to choose 4 cups out of 6 cups of two samples (2 samples*3 cups each = 6 cups);
$$C^4_9$$ - total # of ways to choose 4 cups out of 9.

Another way:

Calculate the probability of opposite event and subtract this value from 1.

Opposite event is that contestant will taste ALL 3 samples, so contestant should taste 2 cups of one sample and 1 cup from each of 2 other samples (2-1-1).

$$C^1_3$$ - # of ways to choose the sample which will provide with 2 cups;
$$C^2_3$$ - # of ways to chose these 2 cups from the chosen sample;
$$C^1_3$$ - # of ways to chose 1 cup out of 3 from second sample;
$$C^1_3$$ - # of ways to chose 1 cup out of 3 from third sample;
$$C^4_9$$ - total # of ways to choose 4 cups out of 9.

$$P=1-\frac{C^1_3*C^2_3*C^1_3*C^1_3}{C^4_9}=1-\frac{9}{14}=\frac{5}{14}$$.

Hope it's clear.
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15 Nov 2009, 03:16
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I got B: 5/14 as well, although using a different approach.

Number of ways to choose 4 cups to drink from: 9C4

Since the contestant must drink 4 cups of tea, and there are only 3 cups of each tea, the contestant MUST drink at least two different samples.

Number of ways to choose which two samples the contestant drinks (since he can't drink all 3): 3C2

Three cases:

a) 3 Cups of Sample 1, 1 Cup of Sample 2: 3C3 * 3C1
b) 2 Cups of Sample 1, 2 Cups of Sample 2: 3C2 * 3C2
c) 1 Cup of Sample 1, 3 Cups of Sample 3: 3C1 * 3C3

Therefore,

$$P = \frac{3C2(3C3*3C1 + 3C2*3C2 + 3C1*3C3)}{9C4}$$

$$P = \frac{3(1*3 + 3*3 + 3*1)}{\frac{9*8*7*6}{4*3*2*1}}$$

$$P = \frac{45}{{9*2*7}}$$

$$P = \frac{5}{14}$$
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15 Aug 2010, 16:17
I thought the answer to this question can be obtained thus: 1 - P(all samples are tasted)

P(all samples tasted) = I choose one from each of the 3 samples = 3C1 x 3C1x 3C1 x 6C1 as there will be 6 cups left from which I can get the 4th cup = 27 x6 = 162 makes no sense as there are only 9C4 combinations = 126.

S1 A B C
S2 D E F
S3 G H I

Select 1 from S1 in 3 ways, same with S2, same with S3 = a total of 27 ways
I have 6 cups left, can select 1 in 6 ways

So total of 27 x 6 = 162 ways. It seems that this should be 81, I am double counting somewhere. Only 81 gives me the right answer...

Ok so where did my brain get fried?
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15 Aug 2010, 17:43
It should be 1-(3*3C2*3C1*3C1/9C4)
= 1-9/14 = 5/14

Now why is your approach wrong ?
E.g: in how many ways 2 objects can be selected from a set of 3.
We know it is 3C2
As per the approach you took above it should be equal to; 3C1*2C1 != 3C2.

Hope it helps.
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15 Aug 2010, 17:50
Can you give the breakdown as to how you arrived at (3*3C2*3C1*3C1). Thanks

My thinking was simply that

-|-|- <- select 1 of 3
-|-|- <- select 1 of 3
-|-|- <- select 1 of 3

That is a total of 27 ways. I can see that these are not unique combinations, but how do I remove the duplicates?
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15 Aug 2010, 18:03
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We want to select 2 cups of tea which are of similar type and the rest 2 cups which each are of different type ;
hence the terms 3C2 * 3C1* 3C1
Now you should multiply it by 3 because ; out of the 3 tea types , you may select the tea type which is in 2 cups, in 3 ways.

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15 Aug 2010, 18:16
Maybe I dont understand the question. But I was approaching it as 1 - p(he tastes all samples), so why limit to just 2 cups and then make it so that the other 2 are each of a different type?

S1 A B C
S2 D E F
S3 G H I

So why can't be go ADGF - select one from S1, one from S2, one from S3 and then have 1 of the remaining 6? You are doing 3c2 3c1 3c1, so that is picking 2 from one row and one each from the other 2 rows such as ABDG. Perhaps if you can explain how to get an accurate count of combinations, when I select 1 from each row, make it 3 cups and then pick the 4th one of the remaining 6?
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15 Aug 2010, 18:53
I am not sure if it could be calculated that way.
Lets make it even simpler
S1 A B C

In how many ways can you select 2 out of these 3 ?
3C2, right ?
If however; if I first select 1 of them say in 3C1 ways; there are 2 still left.
Now, if I again select 1 out of the remaining two, in 2C1 ways, the total no. of combination 3C1*2C1, which is not equal to 3C2.
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15 Aug 2010, 19:06
Yes I can see that clearly I am ending up with more combinations than necessary. I was thinking if there is way to factor out those repeats the way I am doing it. I see your point in the simple example. However when I think of this, I am thinking of a 3x3 matrix where I have to make sure I get one from each row and the 4th can come from anywhere - that is the same thing as saying 2 from a row and 1 each from the remaining 2 (your and correct approach). My count differs from yours by a factor of 2 (I am 162 and you are 81). How can I rationalize this?
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17 Aug 2010, 00:53
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AKProdigy87 wrote:
I got B: 5/14 as well, although using a different approach.

Number of ways to choose 4 cups to drink from: 9C4

Since the contestant must drink 4 cups of tea, and there are only 3 cups of each tea, the contestant MUST drink at least two different samples.

Number of ways to choose which two samples the contestant drinks (since he can't drink all 3): 3C2

Three cases:

a) 3 Cups of Sample 1, 1 Cup of Sample 2: 3C3 * 3C1
b) 2 Cups of Sample 1, 2 Cups of Sample 2: 3C2 * 3C2
c) 1 Cup of Sample 1, 3 Cups of Sample 3: 3C1 * 3C3

Therefore,

$$P = \frac{3C2(3C3*3C1 + 3C2*3C2 + 3C1*3C3)}{9C4}$$

$$P = \frac{3(1*3 + 3*3 + 3*1)}{\frac{9*8*7*6}{4*3*2*1}}$$

$$P = \frac{45}{{9*2*7}}$$

$$P = \frac{5}{14}$$

I don't follow this answer. (a) and (c) show that he ends up tasting all the cups of the given sample, we are not supposed to allow this to happen?
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31 Aug 2010, 02:18
Nice question... economist always posts quality question... thanks for that...
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21 Sep 2010, 17:20
great explanation Bunuel
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10 Nov 2010, 00:11
I was all over the place in trying to solve this...neways Thanks Bunuel..that was a magnificient and clear solution..but my only concern is on the actual GMAT..is this question easy enuf to be answered in under 2 mins...???or rather...it looks like a difficulty level 800+ to me...

+ 1 to Bunuel.....
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i solved the question as follows. i hope my approach is correct.

let's consider that the contestant tasted all 3 flavors.
the first cup can be selected from any 9 cups. P = $$\frac{9}{9}$$
the second cup can be selected from 6 (other 2 flavors) of the remaining 8 cups. P = $$\frac{6}{8}$$
the third cup can be selected from 3 (flavor not tested yet) of the remaining 7 cups. P = $$\frac{3}{7}$$
the fourth cup can be selected from any of the remaining 6 cups. P = $$\frac{6}{6}$$

there are $$2$$ ways in which second & third cup can be tested {AB, BA}. first & fourth cup already have probability 1.

probability that the contestant tasted all flavors = $$\frac{9}{9}*\frac{6}{8}*\frac{3}{7}*\frac{6}{6}*2 = \frac{9}{14}$$

required probability = $$1 - \frac{9}{14} = \frac{5}{14}$$

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03 Aug 2011, 23:47
Can anyone tell me where I went wrong? I understand you can derive the answer from the solution above, but why does my method below give me a wrong answer?

I used 1 - (prob of drinking all three teas)
Total combinations = 9C4 = 126
prob of drinking all threes = 1 of each tea + 1 of any tea.
Using A, B, & C to represent the teas:
Combinations where all three teas are represented (2 of one tea and one of each of the other tea) =
A, A, B, C
A, B, B, C
A, B, C, C

Combinations of rearranging A,A,B,C = 4!/2! = 12
Combinations of rearranging A,B,B,C = 4!/2! = 12
Combinations of rearranging A,B,C,C = 4!/2! = 12

Total combinations of drinking each tea = 36
Probability of not drinking all tea = 1 - (36/126) = 90/126 = 5/7

I can't figure out where the logical error or where I double counted or ignored possibilities come from. Can anyone help?
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26 Aug 2011, 15:59
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perhaps bunuel can chime in / elaborate on dimitri92's answer?

using the 9/9 * 6/8 * 2 * 3/7 * 6/6

this approach always confuses me.
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29 Aug 2011, 17:35
pinchharmonic wrote:
perhaps bunuel can chime in / elaborate on dimitri92's answer?

using the 9/9 * 6/8 * 2 * 3/7 * 6/6

this approach always confuses me.

bump on this? i'm curious about the 6/8 and the factor of 2.

9/9 i understand, but 6/8 should give the probability of either two types, then why is there another factor of 2?
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22 Dec 2011, 22:41
Bunuel wrote:
Economist wrote:
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?

# $$\frac{1}{12}$$
# $$\frac{5}{14}$$
# $$\frac{4}{9}$$
# $$\frac{1}{2}$$
# $$\frac{2}{3}$$

And the good one again. +1 to Economist.

"The probability that a contestant does not taste all of the samples" means that contestant tastes only 2 samples of tea (one sample is not possible as contestant tastes 4 cups>3 of each kind).

$$\frac{C^2_3*C^4_6}{C^4_9}=\frac{5}{14}$$.

$$C^2_3$$ - # of ways to choose which 2 samples will be tasted;
$$C^4_6$$ - # of ways to choose 4 cups out of 6 cups of two samples (2 samples*3 cups each = 6 cups);
$$C^4_9$$ - total # of ways to choose 4 cups out of 9.

Another way:

Calculate the probability of opposite event and subtract this value from 1.

Opposite event is that contestant will taste ALL 3 samples, so contestant should taste 2 cups of one sample and 1 cup from each of 2 other samples (2-1-1).

$$C^1_3$$ - # of ways to choose the sample which will provide with 2 cups;
$$C^2_3$$ - # of ways to chose these 2 cups from the chosen sample;
$$C^1_3$$ - # of ways to chose 1 cup out of 3 from second sample;
$$C^1_3$$ - # of ways to chose 1 cup out of 3 from third sample;
$$C^4_9$$ - total # of ways to choose 4 cups out of 9.

$$P=1-\frac{C^1_3*C^2_3*C^1_3*C^1_3}{C^4_9}=1-\frac{9}{14}=\frac{5}{14}$$.

Hope it's clear.

Thank you.
If the question was about the probability that a contestant does not taste all the 3 samples of a cup. (That's what I wrongly understood from this question first). Then the answer would be $$\frac{6}{7}$$?
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Re: At a blind taste competition a contestant is offered 3 cups [#permalink]

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05 Mar 2013, 08:23
Not sure if my channel of thought is flawed to start with, but when it is said that it's a 'blind' taste competition, I am considering repetitions. As the contestant does not know which of the 9 cups he/she had picked up the first time, the same can possibly be repeated in the second turn and so on until the fourth.
That gives a complete different perspective to the problem.
Where am I going wrong here?
Re: At a blind taste competition a contestant is offered 3 cups   [#permalink] 05 Mar 2013, 08:23

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