Bag of Beans


Suppose there is a bag containing exactly two beans, each of which is either white or black. I know nothing about the expected distribution of black and white, only that each bean is one or the other. I withdraw one bean and don't replace it. If this bean is white, what is the chance that the other bean is black? Surely the answer is 1/2.

Suppose that the bag contains not two beans but a million beans, each of which is either white or black. I randomly withdraw 999,999 beans and don't replace them. If all these beans are white, what is the chance that the remaining bean is black? Surely the answer is still 1/2, as the act of withdrawing each bean is an independent event that should have no effect on the probability of the next bean being one colour or another.

Yet something tells us this cannot be right. If you randomly removed 999,999 white beans from a bag containing one million beans, then you would strongly expect the last bean to also be white. For if there were a black bean in the bag then the chance of missing it in 999,999 trials is exactly one in a million. This is because if we reversed the order of picking then we would have to select the sole black bean out of one million beans on the very first trial. These are tough odds.

If we accept the view that the chance of a black bean is one in a million in the million bean case, then what does that say about the chance of a black bean in the two bean case? Is it a little or a lot less than 1/2?

How do we resolve this apparent paradox?

Explanation

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