A bag initially contains one chip, known to be either white or black. A white chip is then put in the bag, and a chip drawn out, which is seen to be white. What is now the probability of drawing a white chip from the bag?

At first sight, it would appear that inasmuch as the state of the bag after the white chip is withdrawn is identical to the initial state of the bag, the probability of drawing a white chip is just what it was initially, namely \(1/2\). This however is not true. Here is my explanation of why.

After adding a white chip to the bag, the possible contents of the bag are:

1. The added white chip (\(W_{2}\)) and a white chip (\(W_{1}\)) that was in the bag initially, or,
2. The added white chip (\(W_{2}\)) and a black chip (\(B\)) that was in the bag initially.

After a white chip is drawn out (which is seen to be white), the possible contents of the bag are:

1. \(W_{2}\) (if the white chip drawn out was \(W_{1}\)), or,
2. \(W_{1}\) (if the white chip drawn out was \(W_{2}\) and the chip initially in the bag was white), or,
3. \(B\) (if the white chip drawn out was \(W_{2}\) and the chip initially in the bag was black).

Therefore the probability of drawing a white chip from the bag on the next draw are \(2\text{ out of }3\).