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Prianka's socks rack contains 11 pink socks, 12 red socks , 13 orange socks, 14 white socks, and 15 brown socks.

How many socks will she have to pull out in the dark to be sure she gets a matching pair ?
Answer using probabilities. Give all details.


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Let us say she picked a pink one in first pull out and a pink one in the second pull out.
A = color of socks in the first pull and B = color of socks in the second pull.

Total number of socks = 65.  When a second socks is pulled out, total number is less by 1 and number of same colored socks in rack is less by 1.

Probability(A=pink and B= pink) = 11/65 * 10/64
P(A=red and B=red) = 12/65*11/64
P(A=orange and B=orange) = 13/65 * 12/64
P(A=white and B=white) = 14/65 * 13/64
P(A=brown and B=brown) = 15/65 * 14/65

Probability of getting a matched pair: sum of all these above = 0.1899


Priyanka pulls out pink socks first, then there are 64 remaining and of them 10 are pink. She has to pull out 54 of the other colours and then the next one she pulls out will be pink. So in the WORST case, she has to pull out a total of 1+54+1 = 56. It is same as (65 – 11 + 2). If she pulls out 56 socks, then there will be a pair of pink socks for sure.

If she pulls out red socks first, then she has to pull out a total of (65 – 12 + 2 ) = 55, in the worst case.

If she pulls out orange socks first, then she has to pull out a total of 65-13+2 = 54, in the worst case.

If she pulls out white socks first, then she has to pull out a total of 65-14+2 = 53.

If she pulls out brown socks first, then she has to pull out a total of 65-15+2 = 52.

On the whole, to be sure that she has a matching pair of socks, she has to pull out 56 socks, in the worst case.

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