The Lottery Paradox

Consider the following lottery problem:

• one million tickets are sold for one dollar each
• one ticket is chosen and the owner of the ticket wins the entire pot

The probability of any one ticket winning is 1/1000000. In other words, the probability of winning is nearly zero but the collective sum of these probabilities is 1.

The paradox involves the rationality of belief. It is rational to believe that one individual ticket may win but the conglomeration of this rational belief leads to the belief that no ticket can win which is clearly a false belief.

The paradox is related to the question of knowledge and the the accumulation of nearly perfect knowledge which leads to a false conclusion. This is not a mathematical paradox in the sense that nearly perfect knowledge of the truth of a set of hypothesis does not even allow one to conclude that even one of the hypotheses hold.

On the one hand, we have mathematical methods which give one result, but on the other hand, we have a philosophical approach which leads to confidence, not certainty, in another result.

A practical example of this problem occurs in digital communication. In a communication channel, like the channels used to distribute web-pages over the internet, there is a small possibility that a digit can be changed during transmission. While the probability is extremely small for any one digit, the conglomeration of millions of digits implies that eventually an error will occur. Internet protocols generate a method for correcting the errors, when they occur, in most cases.