The sleeping beauty problem is a famous problem where a coin is flipped and then a subject is put to sleep. If the coin was heads they will be awoken on Monday asked what their belief is that the probability was heads. If the coin was tails they will also be awoken on Monday and asked the same question. But, they will then be put back to sleep, have their memory erased and reawoken on Tuesday and asked the same question again.
It seems to me with the Sleeping Beauty problem we have the following with h = heads, t = tails, m = Monday, and tu = Tuesday.
$$ \begin{align} P(h) &= P(t) = 1/2 \\ P(m|h) &= 1 \\ P(m|t) &= 1/2 \\ P(tu|t) &= 1/2 \end{align} $$
Since we don't know what day it is, we want to answer the question of $P(h| m \cup tu)$.
But from Bayes rule we have,
$$ \begin{align} P(h| m \cup tu) &= P(m \cup tu | h) P(h) / P(m \cup tu) \\ &= \left(P(m | h) + P(tu | h)\right) P(h) / P(m \cup tu) \\ &= \frac{(1 + 0)\times 1/2}{ 1} \\ &= 1/2 \end{align} $$
Can someone point out the flaw in the above reasoning?
The puzzle raised by the Sleeping Beauty problem is that there are two conflicting intuitions about it.
The halfer intuition is that the subject starts off with a subjective probability that the coin will land heads of 1/2. It continues that on waking up, the subject has not undergone any informative learning experience, so the subject should continue to have the same subjective probability.
The thirder intuition is that you can convince yourself that if the experiment is repeated, in the long run, the coin will have landed heads about 1/3 of the times that the subject is woken up, so that the subject should, on waking, change their subjective probability to 1/3.
You can try to formalize the halfer intuition by using Bayes rule to argue $$P(\mathrm{heads} \,| \,\mathrm{mon} \lor \mathrm{tue})= 1/2$$
But that after all is just a conditional probability. Someone who accepts the thirder intuition will say that in the circumstances of Sleeping Beauty, that conditional probability does not correctly describe what the subject's subjective probabilities should be on waking.
For those who are sympathetic, that leads to a debate about what exactly is going wrong with this application of Bayes rule. This is a specific instance of a general problem about how Bayesians should think about self-locating beliefs; see especially the section on decision theory.