Jim Worthey • Lighting & Color Research • jim@jimworthey.com • 301-977-3551 • 11 Rye Court, Gaithersburg, MD 20878-1901, USA
 "Girl Walks In," a Third Version of the Boy/Girl Puzzle

The third version of the boy/girl puzzle is the most tricky.
In the first two puzzles, we learn that a family has 2 children. We are then given partial information about the children's genders:

Puzzle #1: The older child is a girl.
Puzzle #2: At least one of the two is a girl.

The question then is, what is the probability that the other child is a girl? As we learned on the first puzzle page, version 1 is simpler. In version 1, we know everything about the older child and nothing about the younger one. The gender of the younger child remains an independent event. In version 2, we have a peculiar kind of knowledge. Some observer—say the mother of the children—knows the genders of both, but tells us only part of what she knows.

Now here is puzzle #3: You know that a family has two children. One of the children walks into the room, and it is a girl. What is the probability that the other child is a girl?

— For a hint, please scroll down —

Hint: Puzzle #3 is NOT the same as puzzle #2.

Answer: 1/2 . That is, if one of the children walks in and she is a girl, the probability is 1/2 that the other child is also a girl.

Explanation

We recall that, for puzzle #1 or #2, there are 4 possible cases:

 A B C D Older girl girl boy boy Younger girl boy girl boy

Review Puzzle #1: If you are told that the older child is a girl, then the possibilities are only A and B. The probability of case A (2 girls) is one out of two, or 1/2.

Review Puzzle #2. If you are told that at least one child is a girl, this excludes case D. The possibilities are A, B, and C, and only in case A is the second child a girl. This is 1 case out of 3, so the probability is 1/3.

In the case of puzzle #3, when "one of the children walks in," there is an additional random event by which one of the two children is chosen. Imagine that the child walks in through a door. Outside the door, where we can't see it, is a sign that says "Older child please walk in," or perhaps "Younger child please walk in."

There are now 8 cases, but we can use the same table above. The sign selects a row, chance selects a column, and then we have an event such as "younger child walks in AND it is a boy AND both children are boys."

So, if the sign says "older" and a girl walks in, we must be in column A or column B, and the probability of 2 girls (column A) is 1/2. If the sign says "younger," and a girl walks in, by similar reasoning, column A or column C applies, and again the probability of 2 girls (column A) is 1/2. If the probabilities of "older" and "younger" are each 1/2, then the probability of 2 girls is (1/2)*(1/2)+(1/2)*(1/2) = 1/2 .

To say it another way, there are 8 cases. In 4 of those cases, a girl walks in. But 2 of the 4 cases are in column A. Therefore the chance of column A (both children are girls) is 2/4 = 1/2.

This puzzle can also be treated as an example in Bayesian Probability:

 Bayesian Probability

Event W is that a girl Walks into the room. Event T is that the family has Two girls. Reverend Bayes then teaches that
P(T|W)P(W) = P(W|T)P(T) . Or,

P(T|W) = [P(W|T)P(T)]/P(W) .

Here by definition,
P(T|W) is the probability of T, given that W has occurred.
P(W) is the a priori probability of W.
Here P(W|T) is the probability of W, given that T has occurred.
P(T) is the a priori probability of T.

Well, P(W|T) is the probability that a girl walks in, if there are 2 girls. P(W|T) = 1.
P(T) = a priori probability of 2 girls = 1/4 .
P(W) = a priori probability of a girl walking in = 1/2 .

Therefore P(T|W) = 1*(1/4)/(1/2) = 1/2 .

This is, of course, the same answer as before. The Venn diagram above was a generic diagram to stimulate thinking about Bayesian probability. Now that we have solved the problem, we realize that T is a proper subset of W. That is, if the family has Two girls, a girl will always Walk in. This diagram is still not scaled according to probability but shows T as a proper subset of W:

Credit Where it is Due: In an earlier version of these web pages, I had confused puzzle #2 and puzzle #3, not realizing that they are different. Donald MacLeod questioned what I had said, leading to this additional puzzle page. Thank you, Don.

<<Back to boy-girl puzzles #1 and #2.

 web site designed by Nick Worthey
 More Puzzle Pages
 Boy/Girl Puzzles #1 and #2 Ask Dr. Math: boy or girl? Marilyn is Wrong! www.mathpuzzle.com
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