**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?**

**SOLUTION:**

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

Notice that

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. .