I learned about this problem from a video of „Veritasium“. The story goes like this: A beauty falls asleep. Then a coin is flipped.
- If it shows tail, the beauty will be wakened up on Monday.
- If it shows head, the beauty will be wakened up on Monday, immediately falling asleep and not remembering anything, and then again on Tuesday.
On each wake up, the beauty gets asked what she thinks the probability of the coin coming up tail is.
This is confusing, because it invokes the concept of probability. As I have written repeatedly in this blog, this concept needs a well defined experiment (mathematically a probability space or random variable) as a background to even exist. Without that, probability does not have a meaning. The video claims that there are two sensible ways to answer this question, 1/3 and 1/2 for tail.
Before we get into the more philosophical questions later in the video, let us device experiments that could mimic the procedure. The most obvious one is to throw a a dice and denote its outcome. Obviously, we get 1/2 for tail.
The experiment, that Veritasium is suggesting is writing down the wakeups after each dice throw and look which ones are caused by tail. If we do that we get 1/3 for tail, because we effectively have counted each head twice.
This is so simple and straightforward that each video or article about the paradox has to add even more confusion to get a story. One example is in the video. The coin flips are replaced by a football match (win for Monday, loss for the Monday and Tuesday), and the wakeup takes place a million times. The beauty is asked on each wakeup how she thinks the match ended. With the 1/3 logic, she could simply give the wrong answer a million times, reasoning that it is a million times „more likely“ that she wakes up on any of the million days after a loss.
It gets worse. The video assumes we are living in a simulation, where someone created a million copies. I find this question so absurd that I won’t go into it.
But there is one lesson to learn from all that. The fact that an event repeats does not make the cause more likely. It could just be that the same cause causes all the repetitions. Yes, the repetitions are a confirmation of the cause. But they do not make the cause more likely to happen. E.g., if all cars today are honking at you, the reason may be that one of your lights fails. At other days, only one car honks. But that does not make the failure of a light more likely.