The laws of probability, so true in general, so fallacious in particular.
A puzzle has recently been brought to my attention. It goes like this: “I have two children. One of them is a boy born on a Tuesday. What is the probability that I have two boys?” This puzzle was posed to an audience at the Gathering for Gardner meeting in Atlanta, Georgia, by Gary Foshee. He went on to say, “The first thing you think is ‘What has Tuesday got to do with it?’ Well, it has everything to do with it.”
I disagree. The day the boy was born is utterly irrelevant. Perhaps it would be best to look at the “accepted” solution from New Scientist before I tell you why it’s hogwash. Its main point is:
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.
Wrong already. The probability of two boys is not 1/3, but 1/2. The reason for this is, to me, glaringly obvious. The possible combinations are not BB, GG, BG and GB, because GB and BG are, in the context of this problem, identical. The order in which the children are born is irrelevant. The possibilities are therefore BB, GG and one-of-each. We can, indeed, eliminate GG because we already have one boy, so that just leaves an equal probability of two boys or one-of-each, or a 50-50 chance of two boys.
I’ll simplify further: if I tell you I flipped a coin twice, and one of the results was heads, what is the probability that I flipped two heads? I could obfuscate the problem by telling you that the head on the coin belonged to Tiberius, and that there was a buffalo on the tails side, and that the rim was embossed with a floral pattern, but none of those things would be relevant if they did not alter the balance of the coin. The chances of my having flipped two heads would be 1/2.
Similarly, the day the boy was born on is irrelevant obfuscation, as is the order in which the children were born. Here is why the New Scientist thinks the day is relevant (note that the final solution they arrive at is an approximation of the solution I arrived at without having to twist myself into a mental pretzel):
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.
Yes, 1/3 was wrong in the first place, and 13/27 is a quite close approximation to the correct answer, which is 1/2.
I rest my case.
Grumpy Old Man by Mark Widdicombe is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.5 License.