Statement of the problem
God flips a fair coin. If heads, he creates one person with a red jacket. If tails, he creates one person with a red jacket, and a million people with blue jackets.
- Darkness: God keeps the lights in all the rooms off. You wake up in darkness and can't see your jacket. What should your credence be on heads?
Solution
Let us first write down the probabilities that the events of our sample space occur.
- The coin is fair. Therefore
$P(H)=P(T)=0.5$ . - If the result of the coin flip is heads, my jacket is sure to be red. Formally,
$P(R|H)=1$ (and$P(B|H)=0$ ). - If the result of the coin flip is tails, then my jacket is red with probability
$\frac{1}{1'000'001}$ and blue with probability$\frac{1.000.000}{1.000.001}$ . Formally$P(R|T)=\frac{1}{1.000.001}$ and$P(B|T)=\frac{1.000.000}{1.000.001}$ .
This amounts to saying that, with the same probabilty, I could be in one of two worlds:
- I could be in a world inhabited by a single person, myself. In that case, my jacket would be red.
- I could be in a world inhabited by 1.000.0001 people. In that case, my jack could be either red or blue.
But does the color really play a role here? After all, the room is dark and I can't see. All I know is that I have a jacket on.
What I can do is to compute
Since no information on the color is available, my estimate coincides with the probability of getting heads.
As far as I call tell right now, this makes sense. I have to let it go through my ahead again a couple of times.