The Sleep Experiment

To make a quick buck, you volunteer for a strange experiment. The details of the experiment are explained to you before you start.

You arrive on a Sunday and are put to sleep. A fair coin is tossed to decide what happens next.

• If the coin comes up heads, you're awakened on Monday only and the experiment ends there.
• If the coin comes up tails, you're awakened on Monday, put back to sleep with a pill, and awakened again on Tuesday. That pill also erases the memory of your last awakening.

This means that whenever you are woken, you do not know what day it is. And each time, the researchers ask you the same question:
“What do you now say is the probability that the coin landed heads?”

So if you are partaking in the experiment and being awakened with that question... what is your answer?

There are actually two popular positions on this one, and mathematicians are still fighting over who is right. Let's find out which side you are on.

If you are a ‘thirder’, then you would answer “33⅓ percent”. See, there's an equal chance that you are awakened when:

• the coin landed heads and it is Monday;
• the coin landed tails and it is Monday;
• the coin landed tails and it is Tuesday.

The question is only about the first one out of these three potential awakenings because that is the only case where the coin landed heads. Thus a probability of one third!

This reasoning seems sound. Yet if you are a ‘halfer’, something will be bugging you.
Before the experiment starts, what would you say the chance is that the coin ends up heads? Easy, that's 50%. Yet at the time you are woken up, no relevant additional information is known to you! So why would the probability be anything other than “50 percent” still?

Both positions make sense. Weird!