The Lottery Paradox and follow up Math puzzle

This is better stated in Wikipedia:

The lottery paradox arises from Henry E. Kyburg Jr. considering a fair 1,000-ticket lottery that has exactly one winning ticket. If that much is known about the execution of the lottery, it is then rational to accept that some ticket will win.

Suppose that an event is considered "very likely" only if the probability of it occurring is greater than 0.99. On those grounds, it is presumed to be rational to accept the proposition that ticket 1 of the lottery will not win. Since the lottery is fair, it is rational to accept that ticket 2 will not win either. Indeed, it is rational to accept for any individual ticket i of the lottery that ticket i will not win. However, accepting that ticket 1 will not win, accepting that ticket 2 will not win, and so on until accepting that ticket 1,000 will not win entails that it is rational to accept that no ticket will win, which entails that it is rational to accept the contradictory proposition that one ticket wins and no ticket wins.

The lottery paradox was designed to demonstrate that three attractive principles governing rational acceptance lead to contradiction:

It is rational to accept a proposition that is very likely true.
It is irrational to accept a proposition that is known to be inconsistent and is jointly inconsistent.
If it is rational to accept a proposition A and it is rational to accept another proposition A', it is rational to accept A and A'.
The paradox remains of continuing interest because it raises several issues at the foundations of knowledge representation and uncertain reasoning: the relationships between fallibility, corrigible belief and logical consequence; the roles that consistency, statistical evidence and probability play in belief fixation; the precise normative force that logical and probabilistic consistency have on rational belief.

History
Although the first published statement of the lottery paradox appears in Kyburg's 1961 Probability and the Logic of Rational Belief, the first formulation of the paradox appears in his "Probability and Randomness", a paper delivered at the 1959 meeting of the Association for Symbolic Logic, and the 1960 International Congress for the History and Philosophy of Science, but published in the journal Theoria in 1963. This paper is reprinted in Kyburg (1987).

Mister Cat objects:

If we insist there IS a winner, then it a raffle, not a lottery. Given that we call it a lottery, there arises the possibility that there is no winner, assuming people choose numbers independently and there will, most likely, be repeats. But justification ('rational to accept') is not well defined in the paradox - like claiming that a 99% chance defines justification - why not a 98% chance? Why not any chance at all? So it seems to me that we are 'justified' in claiming that there is a winner in a true lottery with 1000 possible selections if the probability that someone wins is greater than the probability that someone does not win, ie, a probability greater than 50%. Of course, the probability that there is a winner may be justified, but still not a true statement. Anyway, given a lottery, or raffle if you prefer, with tickets numbered from 1 - 1000, and only one winning number, how many tickets must be purchased to yield a probability > 1/2 that there is a winning ticket?
(meow)

I think it is rational to believe that any individual ticket will not win, without arriving at an obvious contradiction. With each successive losing ticket, we adjust our probabilities based on the given information and we slowly go from believing the ticket is not a winner, to believing it must be one. I understand the thought exercise, but it seems like a cross between Xeno's Paradox and Monty Hall problem, both of which are more counterintuitive than contradictory.

"accepting that ticket 1 will not win, accepting that ticket 2 will not win, and so on until accepting that ticket 1,000 will not win entails that it is rational to accept that no ticket will win"

is the statement I don't accept.

There's no reason "probably not X" and "probably not Y" implies "probably not (X or Y)", for any definition of "probably not". (Except where "probably not" actually means "definitely not".)

The paradox is not mine - I quoted from Wikipedia, and it apparently goes back to the 1950's. I read about it in a philosophy book (not a Math or Stat book). For what it's worth, I agree with your objection stated, pokes. And as for the Math puzzle I offered, the correct solution is 693.
mc