Compute the Probability

Question

A drawer contains 16 black socks and 6 white socks. Two socks are selected at random from the drawer without replacement. compute the probability that both socks are the same color?

Answer 1

There are 16+6=22 total socks. The number of combinations of 2 of them is

22!/(20!*2!)

Each combination is equally likely to be chosen.

The number of combinations where both socks are white is

6!/(4!*2!)

 

The number of combinations where both socks are black is

 

16!/(14!*2!)

 

The number of combinations where both are the same color is the sum of these two numbers:

 

6!/(4!*2!)+16!/(14!*2!)

 

The probability that both socks are the same color is this number divided by the number of all possible combinations:

(6!/(4!*2!)+16!/(14!*2!))/(22!/(20!*2!))=((6!)/(4!)+(16!)/(14!))/((22!)/(20!))=(6*5+15*16)/(22*21)=(5+5*8)/(11*7)=45/77=.584