First put forth by Henry Kyburg in 1961. It tries to demonstrate that three principles of rational acceptance are logically inconsistent. These principles are:
  1. If it is very likely that a certain conclusion is true, then it is rational to accept that conclusion.
  2. If it is rational to accept that p is the case, and it is rational to accept that q is the case, than it is rational to accept that both p and q are the case.
  3. It is never rational to accept propositions which you realize to be inconsistent.

And now, the paradox:

Suppose that there is a lottery in which 100,000 tickets are sold (and only one will win). The probability that any one given ticket will win is very low -- 0.00001. Therefore, by principle 1, it is rational to believe that my ticket, #1, will lose. And it is also rational to believe that ticket #2 will lose. And #3, and #4, and #5, .... and #99998, and #99,999, and #100,000. According to principle 2 above, it is rational to assume that all 100,000 tickets will lose. But as already stated, one ticket will win.

The solution? Deny principle 1, 2, or 3. (Or engage in some tricky philosophical judo. That's good too.)

Have fun!