Solution to the sleeping beauty problem: 1/2.

Why?

              HEADS  monday (A)
            /
 coin flip <
            \         
              TAILS  monday (B)  tuesday (C)

A, B and C are NOT independent cases as it might seem and suggest the 1/3
answer, because cases B and C aren't independent: if B happens, C happens as
well and vice versa, if C happens, B happens as well. So B and C are in fact a
single outcome. Whether we are awoken at B or C is just a subcase of TAILS and
is irrelevant to reasoning about the coin flip. The credences are therefore:

A = HEADS = 1/2
B = C = 1/4 = TALIS / 2
TAILS = B + C = 1/2
HEADS + TAILS = 1

Imagine the modified case in which TAILS awakes the bauty a million times. Would
this make the probability of coin landing HEADS 1/1000001? Certainly not. There
will be a million more cases of awakening for TALIS, but we can imagine these
happening simultaneously, as a single awakening, since all of the 1000000
awakenings are 100% correlated. The probability of correctly guessing the
sequence number of awakening in the TAILS case will be 1/1000000 but that's a
completely different question, the original question asks about the coin flip
credence, which is 1/2.