I think the relevant question even in your example of just one lottery is to consider the probability that Juliet winning is small. And there's no problem in that. The probability of Juliet winning given that it was rigged is 1. The probability of Juliet winning normally is very small.

But that's fine. That's the probability of Juliet winning given chance, not the probability of chance given Juliet winning. In order to arrive at the second, we need to look at prior probabilities. Given that the rate at which lotteries is rigged is probably less than the chance of Juliet winning, one would still conclude Juliet won fairly.

An example of something that makes this clear is imagining that every other lottery is rigged. If every other lottery is rigged, would it make sense to look at the probability of merely "someone" winning the lottery. Clearly not. Since the probability of "someone" winning the lottery is very high if not 1 depending on the lottery. But does that mean the probability of it occurring by chance is close to 1? No. Because every other lottery is still rigged.