I just came back from the DICE2014 conference and as I recover from the jet lag and prepare posts about the conference I'll present the last topic in the statistics mini-series, the sleeping beauty problem. Unlike the Monty Hall problem, there is no consensus on the right solution even among the experts which makes this problem that much more interesting.
So here is the setting: Sleeping Beauty participates in an experiment. Every day she is explained the process and she is asked about the the credence (degree of belief) that a certain fair coin landed heads or tails. So what is the big deal you may ask? The coin is fair and this means that it lands heads 50% of the time, and tails 50% of the time. However, there is a clever catch.
Whenever the sleeping beauty is put to sleep she takes an amnesia drug which erase all her prior memory. If the coins lands tails, she will be woken up Monday and Tuesday but if the coins lands heads, she will be woken up only Monday. On Wednesday the experiment ends.
So now for the majority opinion: the thirders:
To make this very clear, let's change the experiment and keep waking up the Sleeping Beauty a million time if the coin lands tails, and only once if the coin lands heads. If she is woken up on a day at random, the chances are really small that she hit the jackpot and was woken up in the one and only Monday. So being woken up more times when the coin lands tails, means that the in the original problem the credence the coin landed heads should be one third. If you play this game many times and attach a payout for the correct guess, you maximize the payout overall if your credence is one third.
Now for the opposing minority point of view: the halfers:
On Sunday, the credence is obviously 50-50 because the coin is fair. Even the thirders agree with this. However, From Sunday to Monday, no new information is gained, therefore the credence should be unchanged and the overall credence should remain 50% throughout the experiment. If you adopt the thirder position you should explain how does the credence change if no new information is injected into the problem.
So which position would you take? There were all sorts of approaches to convince the other side, but no-one had succeeded so far.