> "given this winning streak in particular, what's the probability of him cheating in this set of games"
I think the problem lies in the antecedent. Given all chess tournaments played, how often would we observe such a winning streak on average? If the number of winning streaks is near the average, we have no indication of cheating. If it is considerably lower or higher, some people were cheating (when lower, than the opponents).
Then the question is, whether the numbers of winning streaks of one person are unusually high. If we would for example expect aprox. 10 winning streaks, but observe 100, we can conclude that aprox. 90 were cheating. The problem with this is that the more people cheat, the more likely we are to suspect an honest person of cheating as well.
Again, this would be different if the number of winning streaks for a particular person were unusually high.
I think the problem lies in the antecedent. Given all chess tournaments played, how often would we observe such a winning streak on average? If the number of winning streaks is near the average, we have no indication of cheating. If it is considerably lower or higher, some people were cheating (when lower, than the opponents).
Then the question is, whether the numbers of winning streaks of one person are unusually high. If we would for example expect aprox. 10 winning streaks, but observe 100, we can conclude that aprox. 90 were cheating. The problem with this is that the more people cheat, the more likely we are to suspect an honest person of cheating as well.
Again, this would be different if the number of winning streaks for a particular person were unusually high.