Suppose you have a drawer with B black socks and W white socks inside it (both B

Question

Suppose you have a drawer with B black socks and W white socks inside it (both B and W are greater than 0). Now, you draw out a sock at random and place it on the bed. Then you draw out a second sock at random and place it next to the first. Write an expression for the probability that both socks are black. What is the smallest possible number of socks in the drawer for which the probability in 6a is 0.5? Suppose we know that there are an even number of white socks. Now what is the smallest number of socks in the drawer for which the probability in 6a is 0.5?

Explanation / Answer

Here is the solution for 6a.

6a.

The number of black sock in the draw: B.

The number of white sock in the draw: W.

The total number of socks in the draw: B + W.

Selecting a black sock at this moment:

Favorable cases: BC1 = B.

Total cases = B + W. So, the probability is: B / (B + W).

Now, selecting another black sock from the remaining socks:

Favorable cases: (B-1)C1 = B-1.

Total cases: B + W - 1. So, the probability is: (B-1) / (B + W - 1)

And combining both this will lead to: B*(B-1) / ((B + W)*(B + W - 1)).

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