One way to think about the game is adversarially. That is, we first have to publicly reveal our distribution for R, and then the opponent arbitrarily chooses A

One way to think about the game is adversarially. That is, we first have to publicly reveal our distribution for R, and then the opponent arbitrarily chooses A and B. If our distribution is "always choose 0" or "uniformly choose an integer between 1 and 100", then the opponent can always pick A and B so that R will never fall between them. But if we pick with a distribution that is positive everywhere (like the normal distribution), it now becomes impossible for the opponent to choose A and B in a way that reduces our chance to zero. She can make that chance arbitrarily small, but it will always be positive.