Jim Worthey, Lighting and Color Research
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"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.

For the answer, please keep scrolling

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.


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


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
generic Venn diagram

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:

Venn diagram showing proper subset

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.

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Nick Worthey
More Puzzle Pages

Boy/Girl Puzzles #1 and #2

Ask Dr. Math: boy or girl?

Marilyn is Wrong!


Basic Facts, New Ideas, etc.

Read a discussion of Amplitude for Color Matching, and see a colorized 3D graph of the spectrum locus graphed in the orthonormal color space

Seek some basic facts?
Color Rendering Basic Facts

30 New Ideas from the two color rendering articles

Who is Jim Worthey?
Read a short biography

Jim's Past Publications?
See list of articles on color, lighting, etc.

Read a version of Richard Feynman's talk on cargo cult science

Items of Interest

Medimmune Flumist

Who is Nick Worthey?
My son, Nick J. Worthey, is an illustrator and graphic designer. You may enjoy his web page, http://www.nickworthey.com. I most enjoy Nick's black and white cartoons.

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Page last modified, 2006 June 17, 13:17