“I have two children. One of them is a boy born on a Tuesday. What is the probability that I have two boys?”
Spoiler Alert! Stop now if you want to figure this out on your own.
The answer is thirteen twenty-sevenths. My gut reaction to the solution was to question how the day of the week that one child is born can influence the probability of the other child being a boy. That was incorrect reasoning because the information, “One of them is a boy born on a Tuesday”, is not information about one of the children. It is information about the pair.
In other words, it is not equivalent to say:
“I have two children. Child #1 is a boy born on a Tuesday. What is the probability that child #2 is a boy?”
The answer to this altered version is clearly 50%. The difference being that the original puzzle does not give certainty that either child in particular is a boy born on a Tuesday. Try thinking through the puzzle starting with the boy born on a Tuesday as a given. Then cycle through all the possible combinations of genders and days of week for the other child. When you get to “boy born on a Tuesday” for the other child you can no longer take “boy born on a Tuesday” as a given for the first child. The logic in the previous sentence is easy to miss when thinking through the puzzle this way. If you do miss this fact, then you are left with the non-equivalent altered puzzle and incorrectly conclude that the answer is 50%.
A better way to think through this is to start by splitting the universe into equally probably parts; the universe being the set of people with two children. For each child there are 14 possibilities (2 genders times 7 days of the week). For each of those possibilities there are 14 possibilities for the other child. That means there are 14 times 14 total possibilities which equals 196 equally probable combinations. If we limit these 196 combinations to just those with at least one of the children being a boy born on a Tuesday, we are left with 27 equally probable combinations. Of those, 13 have two boys; therefore, the probability is 13/27.
If you change the puzzle to, "I have two children. One of them is a boy. What is the probability that I have two boys?", then answer becomes 1/3. The universe of two-child families is [(boy,boy), (boy,girl), (girl,boy), (girl,girl)]. We learn that at least one child is a boy which excludes (girl,girl). We are left with three equally probable combinations; one of which is two boys. 1/3.
If you want to read a robust debate and discussion of this puzzle, check out this Skeptics Guide to this Universe forums thread.
I'll finish with my T-SQL solution:
WITH child AS
(
SELECT g.gender, w.weekday_born
FROM
(VALUES('boy'),('girl')) AS g(gender)
CROSS APPLY
(VALUES('Sun'),('Mon'),('Tue'),('Wed'),('Thu'),('Fri'),('Sat'))
AS w(weekday_born)
) SELECT
CAST(SUM(CASE
WHEN c1.gender = 'boy' AND c2.gender = 'boy'
THEN 1 ELSE 0 END) AS FLOAT)
/ COUNT(*) AS answer FROM
child AS c1 CROSS JOIN
child AS c2 WHERE
(c1.gender = 'boy' AND c1.weekday_born = 'Tue')
OR
(c2.gender = 'boy' AND c2.weekday_born = 'Tue')/*
answer
----------------------
0.481481481481481
*/
3 comments:
Thank you. After reading your "splitting the universe" paragraph, I was finally able to visualize this problem and feel satisfied with the answer. It's amazing how subtle probabilities can be.
Bonus points for the sql solution.
This problem seems to be making comeback. Gary Foshee introduced it at a 2010 conference honoring Martin Gardner, the long-time author of Scientific American's "Mathematical Games" column. His answer was wrong, and one Charles Jenkins repeats it. To illustrate, I'm going to change the problem in a couple of ways that have absolutely no impact on how we should get to the answer. I'll start with the simpler version:
You run across an old school buddy - you were both in the Mathematical Games Club - on the street. While getting reacquainted, he tells you "I have two children, and one of them is a XXXXX. Given that information about a gender that applies to one or both children, what is the probability I have a boy and a girl?" But in the middle of his statement, a passing driver honked his horn, drowning out the gender your friend named (as represented by XXXXX). Can you still answer?
Yes, you can, but only if you change the answer to the original question to 1/2.
If you had actually heard your friend say "boy," then Gary Foshee and Charles Jenkins say the answer is 2/3 (one minus the chance of two boys). Gary Foshee says this, because he remembers that Martin Gardner once said it for a similar problem. Many other people believe it, either because they haven't thought it through carefully, or didn’t know that Martin Gardner changed his answer.
If you accept their answer, we have a paradox. Because you would also have to say 2/3 if the obliterated word was "girl." And if the answer is 2/3 regardless of which word was obliterated, it has to be 2/3 even if your friend hadn't made a statement about gender. But we know that probability is 1/2.
This is a variation of a famous problem introduced by Joseph Bertrand in 1889, called Bertrand's Box Paradox. His point, reiterated by Gardner, was that knowledge of the possible combinations is not enough to solve a problem like this. You also need to know why this particular fact was given to you, when other equivalent facts could have been.
If your friend actually has two boys, or two girls, there were no alternatives. But if he has one of each, he had to choose between them. So you can't equate the statement "One is a boy" with the event "I represent all possible combinations that include a boy." Because some parents that have a boy will tell you "One is a girl."
And if you don't know how the decision was, you can only assume a random choice (because otherwise the answer is different for boys and girls). That is, such a parent has a 50% chance to say "boy," and a 50% chance to say "girl." And that means that the answer Foshee and Charles Jenkins should have given is 1/2, not 2/3. Yes, 2/3 of all two-child families at least one with a boy have two, but only half of the fathers who would randomly *tell* you about a boy have two.
And it doesn't matter if other facts are included in the XXXX that you didn't hear. The answer is always 1/2, exactly, no matter what is included. 1/3 is correct only if your friend was forced to tell you about a boy, and could not tell you about a girl. 13/27 is correct only if he had to tell you about Tuesday as well. The answer changes because a two-boy family is almost twice as likely, compared to a one-boy family, to have one born on a Tuesday.
Hi JeffJo. Thanks for the comment. My response wouldn't fit in the comments section so I posted a separate blog entry. Thanks again.
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