Someone comes up to you and tells you he flipped ten coins for ten people. They were fair coins, but only three came up heads. What is the probability yours was heads?

There are three people of ten who got heads. There is a 30% chance that you're one of those three, right?

Now take the sleeping beauty paradox. A coin is flipped. If it lands on heads, the subject is woken twice. If it lands on tails, the subject is woken once. For simplicity, assume it happens exactly once, and there are one trillion person-days. You wake up groggy in the morning, and take a second to remember who you are.

If the coin landed on tails, that would mean that there is a one in a trillion chance that you will remember that you're the subject. If it was heads, it would be two in a trillion. As such,  if you do remember being the subject, the probability that it's heads is P(H|U)=P(U|H)*P(H)/[P(U|H)+P(U|T)] = (2/trillion)*(1/2)/[(2/trillion+1/trillion)] =2/3, where H is coin lands on heads, T is coin lands on tails, and U is you are the subject.

Technically, it would be slightly less than 2/3, since there will be one more person-day if the coin lands on heads.