The Lottery Paradox Versus the Math of Probability


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Who will hold the winning ticket? : Image by Oceanview Med Spa”

The term “Lottery Paradox” usually describes an odd problem that begins with a person’s belief that any one specific raffle ticket will lose.

This belief implies that no ticket will win.

However, if the lottery guarantees that a winning ticket will indeed be drawn, the person should simultaneously believe that some ticket will indeed win.

Therefore the person’s twin beliefs form a paradox.

Some scholars describe this situation as a “Lottery Puzzle,” but I’d like to reserve that term for a different situation.

For now, let’s see how to resolve the Lottery Paradox using the math of probability theory.

The Lottery Paradox Requires the Guarantee of a Winner

To begin the “Lottery Paradox,” a raffle must guarantee that exactly one winner will be drawn from a large number of tickets. For example, when a raffle ticket is sold, a copy is placed into a hat from which the winning ticket will later be drawn.

Assuming that the raffle is fair, each ticket has an equal chance of becoming the winner. For example, if 100 tickets were sold, each ticket has one chance in a hundred, or 1/100 = 1% of being drawn.

A punter holding one raffle ticket would rightly assert his 1% probability of winning. If pressed, however, he may agree that “It is false to state that my ticket will win.

Since each ticket has the same low chance to win, it would be equally correct to say that each ticket will not win.

However, that implies that no ticket would win. The paradox is that the raffle guarantees that exactly one ticket will indeed win.

Epistemology and the Lottery Paradox

The Lottery Paradox is often presented as a problem in epistemology, the theory of knowledge. “What can we truly know about a lottery or any future event tied to probabilities?”

This article concerns mathematics rather than the philosophy of epistemology, so let’s leave the resolution of this question as a challenge to philosophers.

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