Originator:

Henry Kyburg, Jr.

Publication:

Kyburg, H. (1961) "Probability and the Logic of Rational Belief", Wesleyan University Press, Middletown

The Paradox:

There are 1000 tickets in a lottery with only one prize.

Argument 1:

999 tickets will lose and there will be one winning ticket.

Argument 2:

Each ticket has a very small chance of winning (a probality of 0.001) and therefore it is likely that no tickets will win.

Thus two apparently acceptable arguments lead to a paradox.

The Solution:

If we add that the lottery is to be drawn in the month of May then using the same logic as in Argument 2 we could argue:

The chance that the lottery will be drawn on any given day in May is very low (a probality of 0.03) and therefore it is likely that the lottery will not be drawn.

And furthermore let’s say there are 100,000 people interested in buying a ticket in this lottery. Again, using the same logic in argument 2 we could argue:

The chance of being able to buy a ticket in this lottery is very low (a probality of 0.01) and therefore it is very unlikely that anyone would be able to buy a ticket.

The logic behind Argument 2 is obviously faulty because there are lotteries drawn all the time where there is one prize and there is always a winning ticket. So let’s do some math.

If we were to calculate the probality that there will be a winning ticket then we should add the individual probabilities of each ticket. There are 1000 tickets with a probability of 0.001 of wining. 1000 * 0.001 = 1 so we can assert there will be a winning ticket.

Alternatively, say if we were to say that the probality of each ticket winning was so close 0 that we could assume it to be 0 then 1000 * 0 = 0. Low and behold we could assert that no ticket would win!

And there’s the problem – Argument 2 subtly implies changing each tickets chances of wining from 0.001 to 0 (or at least closer to 0).