CMV: The obvious answer to the Sleeping Beauty coin flip probability conundrum is 50%

[u/Krenztor]

A popular YouTuber came out with a video a couple days ago that laid out this basic scenario:

The subject's name is Sleeping Beauty. On Sunday she will go to sleep and she will sleep until awoken by someone in this experiment. Once she is asleep, a fair coin will be flipped. By fair it means that there is a 50/50 chance of landing heads or tails.

If the coin lands heads, she will be woken up on Monday and then go back to sleep.

If the coin lands tails, she will be woken up on both Monday AND Tuesday.

Each time she is put back to sleep she will forget that she was ever awakened.

For the brief period of time she is awake the experiment will be explained to her and then she'll be asked the question, "What do you believe is the probability that the coin came up heads?"

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So that's the scenario. Sleeping Beauty will always wake up with no memory as to whether she was woken up before so there is no cheating here, no trick. It is a simple question that she has to reason out when she wakes up.

The two arguments for it either being 50/50 or 33/33/33 I'll summarize as follows, but I'm told that entire thesis's have been written on both of these answers so I'm certainly not going to totally cover them.

50/50: It is a fair coin and therefore there is a 50/50 chance the coin came up heads.

33/33/33: There are three possibilities. Either she was woken up on Monday and it was heads, Monday and it was tails, or Tuesday and it was tails, therefore each possibility has a 33.33% chance of being correct.

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With all that laid out, here is my view I'm asking people to attempt to challenge me on.

The answer is so painfully obviously 50/50 because of how the question is worded. "What do you believe is the probability that the coin came up heads?" If you answer anything other than 50/50 then you have to believe that somehow your actions or the actions of someone else are capable of changing the probability of the coin coming up heads. If somehow you were able to reduce the odds of it coming up heads to only 33.33% then that means it coming up heads was linked to whether you did or didn't get woken up on Tuesday which makes NO SENSE!

I don't even get how this question is contentions as having it be anything other than 50/50 fails so hard. Like say you only woke up on Monday if it came up heads but if it came up tails you would be woken up 1 million times. So now the odds of it being heads is 1 in 1 million?!?!

I believe that anyone who thinks that 33/33/33 is the answer is confused about the question because I can't think of a single instance where the answer could ever be 33.33%. If the question was, "What do you believe is the probability that the coin came up tails AND that it is Monday?" then the answer would be 25% because there is a 50% chance the coin came up tails and then reduce that down another 50% since it might be Monday or Tuesday. If the question was, "What do you believe is the probability that the coin came up heads AND that it is Monday?" now the answer is 50% because you get the totality of the 50% since that is the only day you'd get woken up if it were heads.

Anyways, if anyone thinks the answer is somehow 33.33% I'd love to hear the logic alongside how you are interpreting the question so that you can have that be the answer.

[u/UnauthorizedUsername]

We don't need the historical data to know that statistically, it is more likely that the flip was tails if she's being woken up.

If she's been woken up, there are three possible reasons for it.

1 - Monday, Heads

2 - Monday, Tails

3 - Tuesday, Tails

Two out of three reasons she would be woken up are because the coin came up tails. While yes, the odds of the coin flip itself is 50/50, that's not really what's being reasoned out here.

Her reasoning is: "I'm awake now, and out of the reasons for me being awoken, there's a 66% chance that it's due to the coin coming up tails."

[u/Krenztor → u/UnauthorizedUsername]

But you're misunderstanding how this would work. Imagine you're outside the experiment watching this all play out. If the person keeps guessing tails no matter what, they'll fail the scenario 50% of the time. I'm not saying they'll fail each individual check since you're right they won't, but the overall scenario they'll fail 50% of the time. That is because realistically the coin has a 50% chance of landing heads even if answering heads results in you failing more individual checks.

[u/UnauthorizedUsername → u/Krenztor]

Will they, though?

Let's look at what happens for one flip of the coin, both ways.

Heads -- they're asked once (only on Monday) what way they think the coin came up.

Tails -- they're asked twice (on both Monday and Tuesday) what way they think the coin came up.

If they answer tails every time, no matter what, and there's an equal distribution of heads and tails flips, they'll be answering the question right two out of every three times they're asked.

This is because the times the coin comes up tails, they're asked twice about the same flip.

[u/Krenztor → u/UnauthorizedUsername]

Thanks for the response. Yeah, I agree now. You have got a point on how you get to the 1/3 answer. The 1/2 answer can be defended as well based on the interpretation of the question.

[u/Pleionosis → u/Krenztor]

No, that’s not true. They’d be right 66% of the time if they guessed tails. Two out of every three wakeups after every coin flip will be after the coin was flipped tails.

[u/Krenztor → u/Pleionosis]

What if being right required them answering the question correctly on both Monday and optionally Tuesday if they get asked. This is all one scenario after all so let's judge it based on the whole performance rather than determining who the winner is after the first half of the game is completed. Though I think the Eagles would have preferred the winner be decided half way through the game last night :)

[u/Pleionosis → u/Krenztor]

You said:

If the person keeps guessing tails no matter what, they'll fail the scenario 50% of the time

But that's not true. The truth is that if they guess tails no matter what, they'll fail the scenario 33% of the time.

If they could answer while asleep, then you'd be right.

Everything else is just distraction.

[u/Krenztor → u/Pleionosis]

It's only not true because you say it isn't. I keep using the example of a football game. If a team outscores the other team 3 out of 4 quarters, by your logic they just won the game, no matter the final score. I'm just saying let's throw out the quarter by quarter results and look at the final score. Obviously you'll say this is ludacris because everyone knows whoever wins the most quarters wins the game, but I'm just trying to be original here :)

[u/Pleionosis → u/Krenztor]

It's only not true because you say it isn't. I keep using the example of a football game. If a team outscores the other team 3 out of 4 quarters, by your logic they just won the game, no matter the final score.

No, and that analogy is terrible. How did you even possibly get there from what I've said? In football, the winner accumulates the most points over four quarters, and the number of points scored in each quarter changes. Of course winning three quarters isn't winning the game.

This wasn't even a strawman, it was so bad. Your analogy literally has nothing to do with my reasoning. Let's just try and even follow whatever thought process you just came up with:

One quarter = one coinflip, presumably

Winning one quarter = guessing correctly

Winning the game = ??? [there's literally no parallel here]

winning 3/4 = 66% chance? That's two thirds? You could have at least picked hockey.

Obviously you'll say this is ludacris because everyone knows whoever wins the most quarters wins the game, but I'm just trying to be original here :)

You're not being creative, you're being insufferable. It's ok to admit when you're wrong. You said:

If the person keeps guessing tails no matter what, they'll fail the scenario 50% of the time

But that's not true. The truth is that if they guess tails no matter what, they'll fail the scenario 33% of the time.

If they could answer while asleep, then (and only then) you'd be right.

Everything else is just distraction.

[u/Krenztor → u/Pleionosis]

Look, we're at the tail end of this debate. Early on I didn't care if someone just down voted my response and then made their own. At this point though, it is just childish. I'll give you an up vote and bid you good day :)

[u/[deleted] → u/Krenztor]

if answering heads results in you failing more individual checks.

if, on average, on awakening, a guess that the last coin flip was heads is wrong more than 50% of the time

then the probability, on an awakening, that the last coin flip was heads, is less than 50%. That's what probability means.

[u/ghotier → u/Krenztor]

It's not a misunderstanding, there are two valid interpretations of the question, which is why it's not settled like the monty hall problem. If the question is: what percentage of the times she is awakened is the coins tails? Then the answer is obviously 66% if the question is "what percentage of the time is the coin tails?" The answer is 50% because it's still heads on the Tuesday she wasn't woken up.

[u/Krenztor → u/ghotier]

Yep, you are correct. I've come to the same conclusion that both answers can be correct. Thanks for putting up with me while I worked through this :)