A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2.
a) How small can the number of socks in the drawer be?
b) How small if the number of black socks is even?
Let red and black socks. The probability of the first sock’s being red is ; and if the first sock is red, the probability of the second’s being red now that a red has been removed is . Then we required the probability that both are red to be , or
Therefore, we can create the inequalities
Taking the square roots, we have, for r>1.
Simplifying, we have
So we can now easily plug in values for b, then solve for r.
When , r must be between 2.414 and 3.414, and so . For ,
Using the same inequality, we can substitute even values for b starting from 2, then solve for the value of r. After, check if the probability is 1/2. Refer to the table below.
So, when b is 6, r is 15 and the probability of getting 2 red socks is 1/2. This condition satisfies the problem. .