Tuesday's Child 
Monday, 21 June, 2010, 04:31 PM - Not TFTD
This problem has been doing the rounds for a little while, including a solution in New Scientist a few weeks ago. I thought I'd offer my thoughts on it to anyone who's interested.

Gary Foshee presented this puzzle at the Gathering for Gardner convention.

"I have 2 children. One is a boy born on a Tuesday. What is the probability I have two boys. The first thing you think is 'What has Tuesday got to do with it?' Well it has everything to do with it."

He then got off the stage, giving no further information and set maths blogs around the world wondering what he was getting at. You might think he's hinting at variations in daily birth rates or male female birth differences, but the solution is actually much more interesting and profound than that.

To begin with lets look at the simpler version of the problem. Assume we know nothing about the day the boy was born. He has two children, one born first one born second and they can be (almost) equally likely of either sex. The possible combination of sexes, first child first, are:

BB BG
GB GG

We know it can't be the last one because he's already told us that one of the children is a boy, so out of the remaining three combinations: BB, BG, GB, only one out of three has two boys in it - so the probability is 1/3.

Now we are told that the boy was born on a Tuesday. What possible difference could that make? On the face of it, the day a child is born shouldn't make any difference. To see why it makes a big difference we need to list all the possible combinations of boys and girls on different days. To simplify things, I'm going to invent a new week with only Mondays and Tuesdays in it. So the possible things that can happen are:

boy born on Mon (BM)
boy born on Tue (BT)
girl born on Mon (GM)
girl born on Tue (GT)

We now draw up a table of all possible combinations of first and second child. There are 16 combinations.


BM+BM BM+BT BM+GM BM+GT
BT+BM BT+BT BT+GM BT+GT
GM+BM GM+BT GM+GM GM+GT
GT+BM GT+BT GT+GM GT+GT


The four on the bottom right consist of 2 girls. So in terms of the easy problem (no mention of days) we now have 12 combinations with at least one boy, of which 4 (top left) contain 2 boys - so we still get 4/12 = 1/3 as the probability of two boys when we don't know that one was born on a Tuesday. But look what happens when we do know that one boy was born on a Tuesday - most of the possible combinations disappear.


----- BM+BT ----- -----
BT+BM BT+BT BT+GM BT+GT
----- GM+BT ----- -----
----- GT+BT ----- -----


We now have only 7 possible combinations of which 3 have two boys in them giving a probability of 3/7 - very different from 1/3.

For a seven day week we just imagine a 14 x 14 table. This gets whittled down to one row and one column that all have a BT event in them - so there are 14+13=27 possible combinations, of which 7+6=13 will be two boys. so the final probability is 13/27 - almost 1/2.

This is a brilliant example of how you have to be very careful when trying to figure out probabilities. Even experts often get them wrong. The "boy born on a Tue" event space is completely different from the "boy born any day" event space.
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