What I don’t get is how P(L(S)) can be bigger than P(W(S)), since I think the former includes the latter. You can only lie about S if S is true, right? So the chance of that happening must be smaller.
No. P(W(S)) is that S won the lottery, and P(L(S)) is that S didn’t win the lottery, but I claimed that S did. P(L(S)) can be bigger than P(W(S)) if S is particularly simple, as all sequences are—approximately—equally likely to win, but simple sequences are more likely to be claimed by liars.
What I don’t get is how P(L(S)) can be bigger than P(W(S)), since I think the former includes the latter. You can only lie about S if S is true, right? So the chance of that happening must be smaller.
>since I think the former includes the latter.
No. P(W(S)) is that S won the lottery, and P(L(S)) is that S didn’t win the lottery, but I claimed that S did. P(L(S)) can be bigger than P(W(S)) if S is particularly simple, as all sequences are—approximately—equally likely to win, but simple sequences are more likely to be claimed by liars.