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# There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yell

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Director
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There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yell [#permalink]

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27 May 2010, 02:41
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There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.
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Re: Probability question: 10 socks [#permalink]

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27 May 2010, 03:59
1
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Expert's post
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nonameee wrote:
There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.

I think you are right.

"Probability that you'll get at least two of the same color" - means at least one pair (one pair or two pairs out of 4 socks).

P(at least one pair)=1-P(no pair) --> no pair means all socks must be of different colors --> $$P=1-\frac{C^4_5*2^4}{C^4_{10}}=\frac{13}{21}$$.

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock;
$$C^4_{10}$$ - total # of ways to choose 4 socks out of 10.

What solution was given in the book for 41/42?
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Director
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Re: Probability question: 10 socks [#permalink]

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27 May 2010, 04:02
Quote:
What solution was given in the book for 41/42?

I think they've forgotten to multiply by 2^4. But the solution was the same you gave.
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Re: Probability question: 10 socks [#permalink]

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28 May 2010, 20:26
Thanks for the explanation.

Bunuel wrote:
nonameee wrote:
There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.

I think you are right.

"Probability that you'll get at least two of the same color" - means at least one pair (one pair or two pairs out of 4 socks).

P(at least one pair)=1-P(no pair) --> no pair means all socks must be of different colors --> $$P=1-\frac{C^4_5*2^4}{C^4_10}=\frac{13}{21}$$.

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock;
$$C^4_{10}$$ - total # of ways to choose 4 socks out of 10.

What solution was given in the book for 41/42?
Intern
Joined: 21 Apr 2010
Posts: 31
GMAT 1: 710 Q48 V40
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Re: Probability question: 10 socks [#permalink]

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28 May 2010, 21:07
Bunuel wrote:
nonameee wrote:
There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.

I think you are right.

"Probability that you'll get at least two of the same color" - means at least one pair (one pair or two pairs out of 4 socks).

P(at least one pair)=1-P(no pair) --> no pair means all socks must be of different colors --> $$P=1-\frac{C^4_5*2^4}{C^4_10}=\frac{13}{21}$$.

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock;
$$C^4_{10}$$ - total # of ways to choose 4 socks out of 10.

What solution was given in the book for 41/42?

wait a sec i dont get it. Why isnt 5 choose 4 $$C^5_4$$ ?
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Re: Probability question: 10 socks [#permalink]

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29 May 2010, 02:31
Detran wrote:

wait a sec i dont get it. Why isnt 5 choose 4 $$C^5_4$$ ?

It's just another way of writing it $$C^5_4$$, $$C^4_5$$, $$5C4$$.
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Re: Probability question: 10 socks [#permalink]

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06 Mar 2014, 10:37
1
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Bunuel wrote:
nonameee wrote:
There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.

I think you are right.

"Probability that you'll get at least two of the same color" - means at least one pair (one pair or two pairs out of 4 socks).

P(at least one pair)=1-P(no pair) --> no pair means all socks must be of different colors --> $$P=1-\frac{C^4_5*2^4}{C^4_10}=\frac{13}{21}$$.

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock;
$$C^4_{10}$$ - total # of ways to choose 4 socks out of 10.

What solution was given in the book for 41/42?

May i knw the logic behind this? m not getting the numerator part ..
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Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !

Math Expert
Joined: 02 Sep 2009
Posts: 36549
Followers: 7078

Kudos [?]: 93146 [0], given: 10553

Re: Probability question: 10 socks [#permalink]

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07 Mar 2014, 00:21
sanjoo wrote:
Bunuel wrote:
nonameee wrote:
There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.

I think you are right.

"Probability that you'll get at least two of the same color" - means at least one pair (one pair or two pairs out of 4 socks).

P(at least one pair)=1-P(no pair) --> no pair means all socks must be of different colors --> $$P=1-\frac{C^4_5*2^4}{C^4_10}=\frac{13}{21}$$.

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock;

$$C^4_{10}$$ - total # of ways to choose 4 socks out of 10.

What solution was given in the book for 41/42?

May i knw the logic behind this? m not getting the numerator part ..

If 4 out of 5 pairs of socks will give one sock, we won't have a matching pair, we'll have 4 different color socks and this is exactly what we are counting in the numerator.

Hope it's clear.
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Re: There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yell [#permalink]

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08 Mar 2014, 04:02
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I solved this one be listing method

Prob of at least getting a pair = 1 -prob(all different at four attempts) = 1-( 1 x 8/9 x 6/8 x 4/7) = 1-8/21= 13/21

a) 1 st attempt = getting any color =1
b) 2nd attempt = not getting the color picked in a). = 8/9
c) 3rd attempt = not getting the two colors above = 6/8
d) not getting any of the four colors above = 4/7
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Re: Probability question: 10 socks [#permalink]

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12 Mar 2014, 10:52
Dear Bunuel

I did 10c1X1c1 X 8c1X6c1/ 10c4 + 10c1X1c1 X 8c1X1c1/10c4.

Thanks & regards
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Re: Probability question: 10 socks [#permalink]

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03 May 2014, 01:00
Why we took 2^4 ?
what if we want to solve it by straight forward method ?
that is considering P ( getting two socks atleast of same color ) ?

Bunuel wrote:
nonameee wrote:
There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.

I think you are right.

"Probability that you'll get at least two of the same color" - means at least one pair (one pair or two pairs out of 4 socks).

P(at least one pair)=1-P(no pair) --> no pair means all socks must be of different colors --> $$P=1-\frac{C^4_5*2^4}{C^4_{10}}=\frac{13}{21}$$.

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock;
$$C^4_{10}$$ - total # of ways to choose 4 socks out of 10.

What solution was given in the book for 41/42?
Math Expert
Joined: 02 Sep 2009
Posts: 36549
Followers: 7078

Kudos [?]: 93146 [0], given: 10553

Re: Probability question: 10 socks [#permalink]

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03 May 2014, 03:37
Expert's post
1
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BOOKMARKED
sunny3011 wrote:
Why we took 2^4 ?
what if we want to solve it by straight forward method ?
that is considering P ( getting two socks atleast of same color ) ?

Bunuel wrote:
nonameee wrote:
There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yellow, 2 green. You select randomly four socks together. What is the probability that you'll get at least two of the same color?

I think it's
[Reveal] Spoiler:
13/21

The book says it's
[Reveal] Spoiler:
41/42
which is in my opinion a mistake.

I think you are right.

"Probability that you'll get at least two of the same color" - means at least one pair (one pair or two pairs out of 4 socks).

P(at least one pair)=1-P(no pair) --> no pair means all socks must be of different colors --> $$P=1-\frac{C^4_5*2^4}{C^4_{10}}=\frac{13}{21}$$.

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock;
$$C^4_{10}$$ - total # of ways to choose 4 socks out of 10.

What solution was given in the book for 41/42?

Can you please tell me which part of below explanation is not clear?

$$C^4_5$$ - # of ways to choose 4 different colors out of 5, basically # of ways to choose which 4 color socks will give us one sock (in this case we obviously will have 4 different colors);
$$2^4$$ - each of 4 chosen colors can give left or right sock.

As for direct approach: it would be lengthier way to get the answer. You should count two cases:
A. one par of matching socks with two socks of different color;
B. two pairs of matching socks.
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Re: There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yell [#permalink]

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24 Jul 2014, 17:07
Total Number of Possible Combination= 10C4= 210

Total number of Combination where socks is not of same colour= 5C4= 5

Probability of Same= 1- 5/210= 41/42

PS: In socks there is nothing as left & right. So the factor of 2^4 is not required.

Hope it is clear & book is right!
Math Expert
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Posts: 36549
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Kudos [?]: 93146 [1] , given: 10553

There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yell [#permalink]

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25 Jul 2014, 02:16
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Expert's post
Total Number of Possible Combination= 10C4= 210

Total number of Combination where socks is not of same colour= 5C4= 5

Probability of Same= 1- 5/210= 41/42

PS: In socks there is nothing as left & right. So the factor of 2^4 is not required.

Hope it is clear & book is right!

The correct answer is 13/21, not 41/42. The point is that each out of 4 color socks you are selecting can give either first or second sock. Please read the whole discussion above.
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There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yell   [#permalink] 25 Jul 2014, 02:16
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# There are 5 pairs of socks: 2 blue, 2 white, 2 black, 2 yell

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