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Why the ‘Sleeping Beauty Problem’ Is Keeping Mathematicians Awake

15 points by RafelMri 3y ago

jmholla 3y ago
I don't get it. The number of times Sleeping Beauty is woken up has no impact on the flip of the coin. The odds are always 1/2, it's just in the tails situation she's asked annoyingly more times.
- carlmr 3y ago
  >The number of times Sleeping Beauty is woken up has no impact on the flip of the coin. But that's not the question. The question is what the probability of heads is, given that she wakes up on a certain day. And she doesn't know what day it is. She's woken up more often if tails occurs. So the conditional probability of tails when she is awoken is higher. Say you repeated this experiment a 100 times. Heads or tails would come up roughly 50:50. Now she would be awoken N_head + 2 * N_tails times. I.e. roughly 150 times. If she always guessed tails she'd be correct 2/3 of the time or roughly 100 times.
  - netsharc 3y ago
    The wording is weird.. "What is the probability of heads?". If they wake her up with a coin toss and ask her "what is the probability of a (totally unrelated) thrown dice landing on the number 6?", the answer to this math problem is 1/6. But in this problem "probability" is related to the results of the coin throws? The wording in the Wikipedia seems better: "What is your credence now for the proposition that the coin landed heads?". Bloody "journalists" ruining everything. Edit: in the original German language article [2] the question is in past tense ("showed" instead of "shows"). Seems to be make more sense. Do I shake my fist at the journalist or the translator now? [1] https://en.wikipedia.org/wiki/Sleeping_Beauty_problem https://en.wikipedia.org/wiki/Sleeping_Beauty_problem [2] https://www.spektrum.de/kolumne/welche-wahrscheinlichkeit-hat-der-muenzwurf-beim-dornroeschen-problem/2125275 https://www.spektrum.de/kolumne/welche-wahrscheinlichkeit-ha...
    - carlmr 3y ago
      It's the awoken part. She's sleeping beauty, and keeps on sleeping if you don't wake her up. The important part is that she only awakes on Mondays or Tuesdays when you actively wake her up. So since she's twice as likely to be woken up when the coin showed tails the conditional probability from her perspective for tails is higher. I don't think the shows or showed part makes the big difference here assuming you left the coin as is after throwing it. It's the conditional probability part. But they did try to explain it in a mysterious click-bait way. I'm pretty sure there aren't many Mathematicians kept awake by this at night.
hsjqllzlfkf 3y ago
Upvote if you're also confused how this problem is distinct from any other conditional probabilities problem from highschool.
cyclotron3k 3y ago
I'm too stupid to see the problem here. It seems that both statements are true and not contradictory: a) the coin flip is 50/50, and b) Sleeping Beauty will see more tails than heads.
bumbledraven 3y ago
Underlying the Sleeping Beauty problem is an Necker-cube-like ambiguity or controversy about whether the denominator of the requested probability is the number of coin flips or the number of awakenings. Arguing about calculations or numbers is pointless until agreement has been reached on this point; correspondingly, once agreement has been reached on the event represented by the denominator, there will be no arguments about calculations because the result is straightforward to calculate.

15 points by RafelMri 3y ago

I don't get it. The number of times Sleeping Beauty is woken up has no impact on the flip of the coin. The odds are always 1/2, it's just in the tails situation she's asked annoyingly more times.

carlmr 3y ago
>The number of times Sleeping Beauty is woken up has no impact on the flip of the coin. But that's not the question. The question is what the probability of heads is, given that she wakes up on a certain day. And she doesn't know what day it is. She's woken up more often if tails occurs. So the conditional probability of tails when she is awoken is higher. Say you repeated this experiment a 100 times. Heads or tails would come up roughly 50:50. Now she would be awoken N_head + 2 * N_tails times. I.e. roughly 150 times. If she always guessed tails she'd be correct 2/3 of the time or roughly 100 times.
- netsharc 3y ago
  The wording is weird.. "What is the probability of heads?". If they wake her up with a coin toss and ask her "what is the probability of a (totally unrelated) thrown dice landing on the number 6?", the answer to this math problem is 1/6. But in this problem "probability" is related to the results of the coin throws? The wording in the Wikipedia seems better: "What is your credence now for the proposition that the coin landed heads?". Bloody "journalists" ruining everything. Edit: in the original German language article [2] the question is in past tense ("showed" instead of "shows"). Seems to be make more sense. Do I shake my fist at the journalist or the translator now? [1] https://en.wikipedia.org/wiki/Sleeping_Beauty_problem https://en.wikipedia.org/wiki/Sleeping_Beauty_problem [2] https://www.spektrum.de/kolumne/welche-wahrscheinlichkeit-hat-der-muenzwurf-beim-dornroeschen-problem/2125275 https://www.spektrum.de/kolumne/welche-wahrscheinlichkeit-ha...
  - carlmr 3y ago
    It's the awoken part. She's sleeping beauty, and keeps on sleeping if you don't wake her up. The important part is that she only awakes on Mondays or Tuesdays when you actively wake her up. So since she's twice as likely to be woken up when the coin showed tails the conditional probability from her perspective for tails is higher. I don't think the shows or showed part makes the big difference here assuming you left the coin as is after throwing it. It's the conditional probability part. But they did try to explain it in a mysterious click-bait way. I'm pretty sure there aren't many Mathematicians kept awake by this at night.

hsjqllzlfkf 3y ago

Upvote if you're also confused how this problem is distinct from any other conditional probabilities problem from highschool.

cyclotron3k 3y ago

I'm too stupid to see the problem here. It seems that both statements are true and not contradictory: a) the coin flip is 50/50, and b) Sleeping Beauty will see more tails than heads.

bumbledraven 3y ago

Underlying the Sleeping Beauty problem is an Necker-cube-like ambiguity or controversy about whether the denominator of the requested probability is the number of coin flips or the number of awakenings. Arguing about calculations or numbers is pointless until agreement has been reached on this point; correspondingly, once agreement has been reached on the event represented by the denominator, there will be no arguments about calculations because the result is straightforward to calculate.