I would frame the question as “What is the probability that you are in heads-space?”, not “What is the probability of heads?”. The probability of heads is 1⁄2, but the probability that I am in heads-space, given I’ve just experiences a wake-up event, is 1⁄3.
The wake-up event is only equally likely on Monday. On Tuesday, the wake-up event is 0%/100%. We don’t know whether it is Tuesday or not, but we know there is some chance of it being Tuesday, because 1⁄3 of wake-up events happen on Tuesday, and we’ve just experienced a wake-up event:
I would frame the question as “What is the probability that you are in heads-space?”, not “What is the probability of heads?”. The probability of heads is 1⁄2, but the probability that I am in heads-space, given I’ve just experiences a wake-up event, is 1⁄3.
The wake-up event is only equally likely on Monday. On Tuesday, the wake-up event is 0%/100%. We don’t know whether it is Tuesday or not, but we know there is some chance of it being Tuesday, because 1⁄3 of wake-up events happen on Tuesday, and we’ve just experienced a wake-up event:
P(Monday|wake-up) = 2⁄3
P(Tuesday|wake-up) = 1⁄3
P(Heads|Tuesday) = 0⁄1
P(Heads|Monday) = 1⁄2
P(Heads|wake-up) = P(Heads|Monday) * P(Monday|wake-up) + P(Heads|Tuesday) * P(Tuesday|wake-up) = 1⁄3