Infinite fair lotteries are well-known to be paradoxical. Let's say that an infinite fair lottery is played twice with tickets 1,2,3,.... Then whatever number wins first, you can be all but perhaps certain that in the next run of the lottery a bigger number will win (since the probability of any particular number winning is zero or infinitesimal, so the probability that the winner is a member of the finite set of numbers smaller than or equal to the first picked number is zero or infinitesimal). So as you keep on playing, you can be completely confident that the next number picked will be bigger than the one you just picked. But intuitively that's not what's going to happen. Or consider this neat paradox. Given the infinite fair lottery, there is a way to change the lottery that makes each ticket infinitely more likely to win. Just run a lottery where the probability of ticket n is 2-n (which is infinitely bigger than the zero or infinitesimal probability in the paradoxical lottery)
What makes the infinite fair lottery paradoxical is that
- there is a countable infinity of tickets
- each ticket has zero or infinitesimal chance of winning.
Suppose now that a past-infinite causal sequence is possible (e.g., my being caused by my parents, their being caused by theirs, and so on ad infinitum). Then the following past-infinite causal sequence is surely possible as well. There is a machine that has always been on an infinite line with positions marked with integers: ...,-3,-2,-1,0,1,2,3,.... Each day, the machine has tossed a fair coin. If the coin was heads, it moved one position to the right on the line (e.g., from 2 to 3) and if it was tails, one position to the left (e.g., from 0 to -1). The machine moved in no other way.
We can think of today's position of the machine as picking out a ticket from a countably infinite lottery. Moreover, this countably infinite lottery is paradoxical. It satisfies (1) by stipulation. And it's not hard to argue that it satisfies (2), because of how random walks thin out probability distributions. (And all we need is finite additivity for the argument.)
So if past-infinite causal sequences are possible, paradoxical lotteries are as well. But paradoxical lotteries are not possible, I say. So past-infinite causal sequences are not possible. So there is an uncaused cause.