On Friday we posed a much harder problem than the previous week, that was first considered in the 1980s, and is still puzzling people.
The problem concerns a sleep experiment, which takes the following form.
The subject takes a sleeping draught on a Sunday and after the subject is asleep a fair coin is tossed.
- If the coin falls as a head the subject is woken on Monday and questioned, and the experiment ends.
- If the coin falls as a tail the subject is woken on Monday and questioned, given more of the sleeping potion, is woken on Tuesday and questioned, and the experiment ends.
The drug that the subject takes also has an amnesiac effect, and so they can't remember if they have been woken before or not.
Each time the subject is woken they are asked "What do you believe the probability of the coin falling as a head to be?"
And the problem that you have to resolve is - what is the correct answer to that question?
There are two schools of thought on this:
Some people hold that as a fair coin was tossed the probability must be a half.
However, others will say that the subject is woken once after a head and twice after a tail, and so they are twice as likely to have been woken after a tail than after a head. Thus the probability is one third.
There is a further theory that the probability from the experimenters' point of view is a half, as 50% of the times they run the experiment the coin will be a head. But from the subject's point of view its one third for the reasons given above.
There is no clear consensus, although it would seem the "thirders" are slightly ahead of the "halfers". You can find plenty of discussion on the internet if you want to explore this further.