Saturday, November 9, 2013

How to interpret probability problems

In The Irksome Tuesday Boy Problem, Rob Eastaway complains of ambiguity in Gary Foshee's infamous puzzle, which asks: "I have two children, (at least) one of whom is a boy born on a Tuesday - what is the probability that both children are boys?"

I don't get the controversy over this. There's no ambiguity. These problems are meant to be thought of as repeated experiments, like surveys. We want to determine P(2 boys | two children at least one of which is a boy born on Tuesday). So this one becomes:

  • Pick a family at random
  • "Hello, sir/madam, do you by chance have exactly two children, at least one of which was a boy born on Tuesday?"
  • If they answer "no", then the interview ends immediately.
  • Otherwise, ask: "Do you have two boys?"
  • If "yes", record this call as a "hit"
  • if "no", record this call as a "miss"

After doing lots of these surveys, let h be the number of hits and let m be the number of misses. The desired probability is then h/(m+h).

If the answer depends on some reasonable assumption that is not explicitly stated in the problem, just calculate the answer based on that assumption and make it explicit it to your answer. Example: "The answer is 13/27, provided that the probability of a newborn being a boy is 1/2 and the probability of being born on Tuesday is 1/7."