A Sleeping Beauty paradox
November 10, 2011 27 Comments
“I’m going to put you to sleep tonight, and wake you up on Monday. Then, out of your sight, I’m going to flip a fair coin. If it lands Heads, I will send you home. If it lands Tails, I’ll put you back to sleep and wake you up again on Tuesday, and then send you home. But I will also, if the coin lands Tails, administer a drug to you while you’re sleeping that will erase your memory of waking up on Monday.”
So when she wakes up, she doesn’t know what day it is, but she does know that the possibilities are:
- It’s Monday, and the coin will land either Heads or Tails.
- It’s Tuesday, and the coin landed Tails.
We can rewrite the possibilities as:
- Heads, Monday
- Tails, Monday
- Tails, Tuesday
I’d argue that since it’s a fair coin, you should place 1/2 probability on the coin being Heads and 1/2 on the coin being Tails. So the probability on (Heads, Monday) should be 1/2. I’d also argue that since Tails means she wakes up once on Monday and once on Tuesday, and since those two wakings are indistinguishable from each other, you should split the remaining 1/2 probability evenly between (Tails, Monday) and (Tails, Tuesday). So you end up with:
- Heads, Monday (P = 1/2)
- Tails, Monday (P = 1/4)
- Tails, Tuesday (P = 1/4)
So, is that the answer? It seems indisputable, right? Not so fast. There’s something troubling about this result. To see what it is, imagine that Beauty is told, upon waking, that it’s Monday. Given that information, what probability should she assign to the coin landing Heads? Well, if you look at the probabilities we’ve assigned to the three scenarios, you’ll see that conditional on it being Monday, Heads is twice as likely as Tails. And why is that so troubling? Because the coin hasn’t been flipped yet. How can Beauty claim that a fair coin is twice as likely to come up Heads as Tails?
Can you figure out what’s wrong with the reasoning in this post?