Tuesday's child is... downright confusing

By Brian Clegg

Having recently revisited the Monty Hall problem, I thought it was worth also taking a look another, arguably even more mind-boggling probability problem that also got Marilyn vos Savant a lot of complaints when she included it in her column.

The problem sounds trivial enough, and comes in the form of a statement for which we have to predict the probability. It reads ‘I have two children. One is a boy born on a Tuesday. What is the probability that I have two boys?’

It sounds trivial. The Tuesday bit is just window dressing, so we are looking at ‘I have two children, one a boy. What is the probability I have two boys?’ So with one child a boy, surely there is 50 per cent chance that the other child is a boy and a 50 per cent chance it’s a girl. Which makes the probability of having two boys 0.5, or 50 per cent. There’s a one in two chance. 

But unfortunately that is not correct.

The reason we get confused is that when trying to imagine the situation we think of the ‘first’ child we come as a boy, then look at the options for the second child being a boy. However the description of the situation would also work if the first child is a girl and the second child is a boy. The only way to be absolutely certain is to work through every possible combination:

These are the four possible combinations, each equally possible. Of these, three are situations that match my initial statement ‘I have two children, one is a boy’. In all but case 4, one of the children is a boy. But only one of those three combinations with a boy also makes the second child a boy. So the answer to ‘I have two children, one a boy. What is the probability I have two boys?’ is not 50 per cent, or one in two, it is one in three. This part of the problem is probably on a par with Monty Hall in the difficulty of getting your head around it. But there is a more fiendish part. We were wrong to discard the Tuesday. Saying the boy was born on a Tuesday changes the probability.

To see this we need a much bigger table. It starts like this:

In total we have 196 entries in this table. We go through every single sex/day combination in the first column combined with a girl born on Monday (fourteen of them in all), then every single sex/day combination in the first column combined with a girl born on Tuesday (fourteen of these too) and so on until we have cycled through every option for the second child.

Now we need to know two things. How many of those pairs feature a boy born on a Tuesday (like option 2 above) and how many of those have a second boy? We are going to have one combination of child A as a boy born on Tuesday with every possible child B – fourteen of those, plus thirteen other combinations where child B was a boy born on Tuesday, but child A wasn’t (we have already counted the instance were both child A and child B are a boy born on Tuesday). So there are 27 rows that match our circumstance of having a boy born on Tuesday.

We now need to pin down how many of those rows had two boys. The first set of fourteen all had a boy as child 1, and half of those – seven also had a boy as child 2. Of the thirteen additional rows where child B was a boy born on a Tuesday, six would have child A also a boy. So of the 27 rows with a boy born on a Tuesday, thirteen of them have a second boy. The answer to ‘I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?’ is 13 in 27 – almost, but not quite, one in two.

This really upsets common sense. Just by specifying the day on which one of the children was born we change the probability of both children being boys from one in three to 13 in 27. Yet our minds rebel at this. Surely we could have chosen any day? The only way I can see to make some sense of this is to point out that in any particular real circumstance, you can’t choose that day at random; it is extra information that depends on the circumstance. The boy will have been born on a particular day and the result of that is that it cuts down the options, just as Monty Hall did when he opened a door and showed a goat. The reality is even harder to accept in this example.

There’s lot’s more on the beguiling nature of probability in my book, Dice World.

Photo by the author

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Now Appearing is the blog of science writer Brian Clegg (www.brianclegg.net), author of Inflight Science, Before the Big Bang and The God Effect.

Source: http://brianclegg.blogspot.com/2025/03/tuesdays-child-is-downright-confusing.html