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In the comments on the linked site several people refer to the Monty Hall problem, which is not relevant in this problem. When I first heard of the Monty Hall problem years ago, I modelled it with playing cards, and by actually doing the experiment I was quickly convinced of the obviousness of the not-so-obvious correct solution. With this problem also, it is very easy to model. Flip a coin. If it is heads, wake up SB on Monday and ask her the odds of flipping heads. It is 50/50. If it is tails, wake her up on Monday and on Tuesday and ask her on both days the odds of flipping heads. It is 50/50. "Thirders" are incorrectly assuming that each of the three outcomes is equally likely. Ain't so. Chance that upon awakening it is Monday is 75% (50% Monday and heads, 25% Monday and tails). Chance upon awakening that is Tuesday is 25%.
The solution at the bottom of the page summed it up very well but basically it boils down to whether it is important that you answer is supposed to be correct for one awakening or one experiment. To answerer correctly one one specific awakening the odds would be 1/3 for heads. To answer correctly for one experiment, it would be 50/50 since the coin is flipped once. Personally I identify as a 'thirder' but as the solution shows https://www.quantamagazine.org/20160129-solution-sleeping-beautys-dilemma/ both are right depending on how the vague initial wording is interpreted.
A very neat puzzle! I actually can't decide which one is correct. One thing I thought of was the idea of quantum superposition: sleeping bauty has not observed the coin toss, so(from her point of view) it is simultaneously heads and tails, so all the three "wakings" are equally real. Given that, I think if sleeping beauty were an electron, she should choose 1/3 because only 1/3 of the equally-real "waking events" invilve a heads coin. But sleeping beauty is not an electron, and the false branch is not real whatsoever, so the analogy kind of breaks down.