Think Nuclear

The “Boy or Girl Paradox” (also called “The Two Child Problem” in addition to other names) is generally phrased as follows:

You know a couple who has two children. At least one of the children is a girl. What is the probability that they have two girls?

This is an ambiguous problem, which leads to different answers depending on the assumptions that are used. Not enough information has been provided to produce a definite answer, and the unstated assumptions fill in the space needed to complete the logic.

Here I investigate this problem and explain the ambiguity.

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I’ve already examined the classic Sleeping Beauty Problem and pointed out some of the pitfalls that many people fail to avoid when trying to solve the problem. I also examined Nick Bostrom’s so-called “Extreme Beauty” modification to the problem, in which Beauty wakes many, many times if the coin toss comes up tails. However, there is another “extreme” variant of this problem, the variant in which the coin toss is replaced with another two-result random process that has extremely uneven odds. That is, in this “extreme” problem, one of the possible results is extremely unlikely. Examining this variant with the methods of reasoning commonly used by the “thirders” can be enlightening and can provide some illustration of why they are wrong.

Since many “thirders” seem to be fond of relying on betting analogies to reason through the problem and explain their arguments, a useful substitute for the coin toss is a lottery. A typical lottery provides a very small chance of winning accompanied by a very large payoff (which is why lotteries are so popular). So here we shall examine what happens when Sleeping Beauty plays the lottery.

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For over 15 years, some people—particularly philosophers—continue to be confused by the so-called “Sleeping Beauty Problem.” This is a rather straight-forward exercise in conditional probability that should be accessible to a student in an undergraduate course on probability and statistics. Nevertheless, there are people who have managed to arrive at the wrong answer to this problem.

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