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

Question

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  13/21

The book says it's  41/42  which is in my opinion a mistake.

Bunuel · Answer

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?

nonameee · Answer

I think they've forgotten to multiply by 2^4. But the solution was the same you gave.

tarun · Answer

Thanks for the explanation.

Detran · Answer

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

Bunuel · Answer

It's just another way of writing it  C^5_4 ,  C^4_5 ,  5C4 .

sanjoo · Answer

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

Bunuel · Answer

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.

gwu · Answer

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

ShantnuMathuria · Answer

Dear Bunuel

I did 10c1X1c1 X 8c1X6c1/ 10c4 + 10c1X1c1 X 8c1X1c1/10c4.
Answer is wrong. Please help!

Thanks & regards

sunny3011 · Answer

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 · Answer

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.

kiranpradhan · Answer

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!

Bunuel · Answer

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

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