Wanna Bet?

The Statistical Mechanic, AKA Wolfgang, had this little puzzle.
A family has two children. We know that one of them is a boy. What is the probability that the other one is a girl?
His answer is:
Initially there are four equally likely cases: { bb, bg, gb, gg}.
The intelligent reader can easily understand what the b and g stands for.
After we learn that there is (at least) one boy, we can eliminate the case gg.
Thus we are left with three equally likely cases {bg, gb, bb} and in two of them
the other child is a girl, thus the probability is 2/3.

I really like this example, because it is so quick and easy and yet so counter-intuitive.


I have an alternative puzzle: We deal out a large deck of cards which consists only of Aces and Jacks, with equal numbers of each. Each hand consists of two cards, face down. I walk up to one at random and turn up one of the cards, getting an Ace. You gentle reader, are offered the following bet: "I will bet $4 to your $6 that the other card is not a Jack, and is in fact another Ace."

Informed by the SM's analysis, should you conclude that the odds are two to one against me, and that hence you should take the bet?

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