As the year draws to a close, our Streuspanne team is getting into a fairytale mood. What exactly is the sleeping beauty paradox and what is actually paradoxical about it? The mathematical probability fairy tale is not about a kiss or a prince, but about a fairytale thought experiment – how could it be otherwise in probability theory – with a coin.
The mathematical probability fairy tale is not about a kiss or a prince, but about a fairy-tale thought experiment – how could it be otherwise in probability theory – with a coin. There are several formulations of this paradox, all of which are very similar. The following setting was chosen for the Streuspanne podcast:
Sleeping Beauty takes part in an experiment. In it, Sleeping Beauty is put to sleep on Sundays and woken up on one or two of the following days. If she is woken up on one day, she is put back to sleep at the end of the day – in such a way that she can no longer remember being woken up. She forgets this waking day and is not told what kind of day it was.
The trick to this experiment is that the waking up takes place according to a certain rule, as the experimenter tosses a fair coin after Sleeping Beauty has fallen asleep for the first time on Sunday. If »heads« falls, she is woken up and questioned on Mondays, but not on Tuesdays. If »Tails« falls, she is woken up and questioned on both Mondays and Tuesdays. A coin toss therefore decides whether she is woken up once or twice. The experiment ends on Wednesday.
The tricky thing about the story is the question that the experimenter ask her during the short waking phases: »What is the probability that the coin came up heads?« Is there even a correct answer here? What influence does the formulation of the experimental situation have on the answer? Where does the self-sampling and self-indication assumption play a role? And what does Sleeping Beauty have to do with our last topics – such as the cube without memory or the end of the world (Doomsday Argument)? Reference is also made to the short episode »Monty Hall Problem (German: Ziegenproblem)«.
In the new podcast episode »Sleeping Beauty Paradox«, our blogging statisticians Sascha Feth and Jochen Fiedler discuss questions relating to the probability paradox in detail with presenter Esther Packullat. The team touches on philosophical thought processes – the anthropic principle, presents new simulation examples and shows what subjective probability means here.
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Fraunhofer Institute for Industrial Mathematics ITWM