The gang at The Browser tease Alex Bellos‘ New Scientist essay “Magic Numbers, Mathemagical Tricksters” thusly:
If the Monty Hall puzzle drove you mad, here’s another one sure to do the same: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”
The piece doesn’t disappoint in this regard.
Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?”
The event was the Gathering for Gardner earlier this year, a convention held every two years in Atlanta, Georgia, uniting mathematicians, magicians and puzzle enthusiasts. The audience was silent as they pondered the question.
“The first thing you think is ‘What has Tuesday got to do with it?’” said Foshee, deadpan. “Well, it has everything to do with it.” And then he stepped down from the stage.
The gathering is the world’s premier celebration of recreational mathematics. Foshee’s “boy born on a Tuesday” problem is a gem of the genre: easy to state, understandable to the layperson, yet with a completely counter-intuitive answer that can leave you with a smile on your face for days. If you have two children, and one is a boy, then the probability of having two boys is significantly different if you supply the extra information that the boy was born on a Tuesday. Don’t believe me?
Seems like nonsense to me, too. I figured, it’s 50/50 and the Tuesday was a distractor. After all, there are only two sexes and it’s 50/50 the second will be a boy. Not so much.
After the gathering ended, Foshee’s Tuesday boy problem became a hotly discussed topic on blogs around the world. The main bone of contention was how to properly interpret the question. The way Foshee meant it is, of all the families with one boy and exactly one other child, what proportion of those families have two boys?
To answer the question you need to first look at all the equally likely combinations of two children it is possible to have: BG, GB, BB or GG. The question states that one child is a boy. So we can eliminate the GG, leaving us with just three options: BG, GB and BB. One out of these three scenarios is BB, so the probability of the two boys is 1/3.
Now we can repeat this technique for the original question. Let’s list the equally likely possibilities of children, together with the days of the week they are born in. Let’s call a boy born on a Tuesday a BTu. Our possible situations are:
- When the first child is a BTu and the second is a girl born on any day of the week: there are seven different possibilities.
- When the first child is a girl born on any day of the week and the second is a BTu: again, there are seven different possibilities.
- When the first child is a BTu and the second is a boy born on any day of the week: again there are seven different possibilities.
- Finally, there is the situation in which the first child is a boy born on any day of the week and the second child is a BTu — and this is where it gets interesting. There are seven different possibilities here too, but one of them — when both boys are born on a Tuesday — has already been counted when we considered the first to be a BTu and the second on any day of the week. So, since we are counting equally likely possibilities, we can only find an extra six possibilities here.
Summing up the totals, there are 7 + 7 + 7 + 6 = 27 different equally likely combinations of children with specified gender and birth day, and 13 of these combinations are two boys. So the answer is 13/27, which is very different from 1/3.
It seems remarkable that the probability of having two boys changes from 1/3 to 13/27 when the birth day of one boy is stated — yet it does, and it’s quite a generous difference at that. In fact, if you repeat the question but specify a trait rarer than 1/7 (the chance of being born on a Tuesday), the closer the probability will approach 1/2.
The bold emphasis is mine, indicating what I consider to be the key to the problem.
Of course, 12/27 is pretty damned close to 1/2. But my reasoning was wrong.
UPDATE: This is actually a more sophisticated variant of Martin Gardner’s “Boy or Girl Paradox,” which dates to 1959.
