Sleeping Beauty Paradox - Numberphile

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Brady hit the nail on the head, "What is the probability the coin was heads?" is a slightly different question from the question "What is the probability that sleeping beauty is woken and the coin was heads?", which is the question that you're always actually asking sleeping beauty, since she has to be awake to ask her.

Asking multiple times without re-flipping the coin doesn't change the probability of flipping the coin, but it changes your probability of getting the answer right.

Numberphile exposure has turned Brady into a bona fide mathematician, and I am firmly here for it.

Can honestly say I never expected to be cast in the role of sleeping beauty in a Numberphile video…

The thirder and halfer arguments are talking about two completely different probabilities: the probability of the coin being heads given that Sleeping Beauty was woken up, and the probability of the coin just being heads. One is conditional, the other is not. Which goes back to what Tom was saying about it being about what we're really asking Sleeping Beauty. Are we asking about the conditional probability or the unconditional probability?

I love when a thought experiment is so strange that you also have to imagine that consent was given

Have you also felt this discomfort when the needle goes into her head?

Like Brady was getting at, it's not a mathematical paradox, but a language paradox because the question is vague enough that it can be interpreted as asking a simple question of the probability of a coin flip, or as asking the likelihood of waking up by a heads or tails. (There was a 66.7% chance you were woken up by a tail, but objectively only a 50% chance that a tail was flipped. Those are two different answers, assuming two different interpretations of the initial question, that don't actually contradict each other)

Brady killed it in this one! It was a very perspicacious way to avoid the paradox.

The third argument is the probability per day. The half argument is the probability per experiment. They're measuring different probabilities. Imagine the experiment is done 10 times, half the time it shows up as a head and half the time it shows up as a tail. And she will always guess a head. Well she will be correct in 10 days out of 30, but she will be correct in 5 experiments out of 10.

With the betting version, if she gets a payout every day, then the thirder stratergy works. If each day she is asked whether she wants to commit to a single bet when she is finally woken up, the halfer stratergy works.

I agree its not a paradox, theres a semantic switch up to conflate probabilities for two different things

Just consider the scenario where she's only woken up if it is a tail and not at all if it is a head. In that case, if she is being asked the question at all, then it means 100% that the experimenter has flipped a tail. Clearly the 1/3 answer is a posterior probability that is completely arbitrary and determined by experiment design. It is the probability that heads HAVE BEEN flipped, like the probability that it HAS rained if you see puddles on the street, which does not tell you anything about how likely it will rain at any given time.

Asking her the question “is it heads or tails?” is different than asking “is her total score for correct vs. incorrect higher if she always answers heads or tails?” Then you realize obviously what she should say.

I feel that there there are two 'probability spaces', one from an external observer and one from sleeping beauty, and these two are squished together. The events from the frame of reference of sleeping beauty is squashed into one half of the outside observer, and the outside observer's space is squashed into the 1/3rd of sleeping beauty.

You can make it more obvious by making two different games: 1) Give her 1 point each time she guesses correctly. This results in the thirder position. 2) Give her 1 point only if she answers correctly every time (note that since she has no recollection of having woken up before she will necessarily guess the same every time). This results in the halfer position.

The confusion arises when you think of every “wake up” being independent, but they aren’t. The 99 wake ups on T is 1 event and the 1 wake up on H is 1 event. There are only 2 events, not 100 events. The answer is 50/50 and the princess gains no more likelihood of being right for saying T. From her pov she only wakes up once and only has one guess.

It’s 50/50. The amount of times you wake her up doesn’t affect the probability. I understand why people think this is a paradox, but it just isn’t. Yes, when you wake beauty up it could be any of the three possible wake up times (two on Monday, one on Tuesday), but two of those wake up times share the same 50% chance of the initial equation.

The question asks about what event happened when the coin was flipped (t or h). When the question is repeated on tuesday it doesnt change that there was only one event that could lead to that. So when they list out P(mon n T) and P(tue n T) they are equal because they are the probability of the same event. Listing both out double counts them and if you count them just once the paradox goes away.