The Lottery Paradox Versus the Math of Probability

The Lottery Paradox Attacks Boolean Logic

This paradox may also highlight the difficulty of extending chains of logic. Suppose we ask the punter, “True or false: ticket number one will win.” After he answers “False,” we repeat the question for each ticket. Boolean logic would say that the raffle’s truth statement is “Ticket one will win OR ticket two will win OR ticket three will win … OR ticket one hundred will win.

George Boole Invented Boolean Logic: Image by Haks via enezeus.com

After the draw, this is a true statement because, although 99 components are false, one component did resolve as true.

Prior to the draw, however, each component was considered false. Therefore the entire Boolean statement is considered false. Yet we know that the raffle guarantees one winner, so the statement should resolve as true.

The Lottery Paradox might even be used to attack Boolean logic because one can justify stating one component but the larger statement becomes misleading.

Probability Resolves the Lottery Paradox

From a mathematics viewpoint, the mathematics of probabilities resolves the Lottery Paradox for a raffle with a guarantee that exactly one winning ticket will be drawn.

The problem started with forcing a 1% chance of winning into a Boolean statement that states, “It is false that this ticket will win.”




Rather, we retain the 1% probability. Since each ticket has a separate, independent number, we can add individual probabilities to calculate the chance that a group of tickets would win.

There is a 50% chance that an odd-numbered ticket will win, if the tickets are numbered and sold consecutively.

Finally, the chance that exactly one ticket will win is the sum of the probabilities that each would win, or 100%.

Lotteries Without Guaranteed Winners

Many lotteries, such as the American Powerball lottery, the Canadian Lotto 6/49 or the UK National Lottery’s Lotto create the winning result regardless of what numbers were guessed by the players. Therefore these lotteries offer no guarantee that any draw will have a winner. Instead, there may be multiple winners or none at all.

Such lotteries present their own puzzle, but that’s a different topic.

References:

Henderson, John R., and Strickland, Jennifer. Boolean Logic. Ithaca College Library. Referenced Dec. 3, 2012.

Hill, Christopher and Schechter, Joshua. Hawthorne’s Lottery Puzzle and the Nature of Belief. Brown University. PDF referenced Dec. 3, 2012.

Sorensen, Roy. Epistemic Paradoxes. (2012). The Stanford Encyclopedia of Philosophy. (Spring 2012 Edition). Edward N. Zalta (ed.). Referenced Dec. 3, 2012.

© Copyright 2012 Mike DeHaan, All rights Reserved. Written For: Decoded Science
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Comments

  1. Joe Edwards says

    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.”

    “If pressed…” Hmmm. That is belief (a forced belief at that) and plays no role into the fact that a single ticket (while it has a small chance) will win. Reality trumps belief.

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

    No. Just, no.

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