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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.