Thursday, February 15, 2007

The Amazing Brain 1: Two quick puzzles

In preparation for a series of posts, I have two puzzles on probability:

1) If someone has two children, and one is a girl, what is the probability that the other is a girl?

2) I have two cards. One is white on one side, and red on the other. The other is red on both sides. If I pull one of them out of a bag, and you only see one side of them - and it's red - what is the probability that the other side is red?

Answers below. Think about it first.


1) You have one known, and one unknown. The sex of the known child has (effectively) no influence on the sex of the other. Thus the probability that the other is a girl is 50%.

2) I'll spell this out painfully to make this clear. In enumerating all possible situations, I'll refer to the seen face first, then the hidden face, then the faces of the hidden card. I'll call the faces R1, R2, R3, and W1. In general situations, when you have two cards where the position matters, you can have eight possible situations:
R1R2 (R3W1)
R1R2 (W1R3)
R2R1 (R3W1)
R2R1 (W1R3)
R3W1 (R1R2)
R3W1 (R2R1)
W1R3 (R1R2)
W1R3 (R2R1)

Now, in the puzzle in question, we can't have the last two, because we know the face up card is red. (In fact, it doesn't matter which way around the card is in the bag, but I enumerated them anyway.) Simply looking at the above set, you can see that there are six possibilities, and the hidden face will be red in 4 of those six, so the probability the hidden face is red is 33 1/3 %.

Did you get both the answers right?

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