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.
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.