You have 7 pairs of socks, all different. Let\'s call them red, orange, yellow,

Question

You have 7 pairs of socks, all different. Let's call them red, orange, yellow, green, blue, purple and violet (because indigo shouldn't be the name of a color.) You put these 14 individual socks in the wash. As you suspected, the dryer does indeed select socks with equal probability and eats them. When the wash is done you only have 11 socks.

What is the probability that you have both red socks?

What is the expected number of pairs of socks that you get back? (You need to have both socks of the same color to count as a pair.) Hint: use linearity of expectations.

Explanation / Answer

Both red socks means there are 12 socks left for washing machine to eat from which is done in:

C(12,3)

Total number of ways for washing machine to eat socks is:C(14,3)

So probability is: C(12,3)/C(14,3)=55/91

Let us look at number of pairs eaten by washing machine

Washing machine eats three socks so at most it can ruin 3 pairs of socks and at least 2 pairs because in worst case all three socks will be of different colours ie belonging to 3 pairs so we get back only 4 pairs and in best case we have 1 pair of socks of one colour eaten and one of other color and we get back 5 pairs

Case 1: All socks eaten of different colour

SO we choose 3 out of 7 colors in :C(7,3). From each of these pairs a sock can be selected in 2 ways since the socks are not identical ,one is for right foot and other for left foot.

So, 2*C(7,3) number of ways

So probability is:

(2*C(7,3))/C(14,3)=5/26

In this case we get back 4 pairs of socks

ie P(X=4)=5/26 where, X is random variable denoting number of pairs of socks we get back

Case 2: 1 pair eaten and one of ther color eaten by washing machine

Choose a pair to be eaten in : C(7,1)=7 ways. Choose another sock from remaining 12 in C(12,1)=12 ways

So probability is: (7*12)/C(14,3)=3/13

IN this case we get back 5 pairs

P(X=5)=3/13

So expectation is:

4*P(X=4)+5P(X=5)

4*5/26+5*3/13=10/13+15/13=25/13

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