For this problem, please ignore the fact that babies are not 50-50 boys-girls. Just assume that half of all babies are boys, the other half girls.

 

A starter question first. You meet two old friends. One says "I have two children. The older one is a boy". The other says "I have two children, too, and at least one of them is a boy." For each friend, what is the probability that both of their children are boys? Hint: The probabilities are different.

 

Now a third old friend joins up with you and says "I also have two children. One is a boy who was born on a Tuesday."

 

What is the probability that both of this third friend's children are boys?

[hide]For the first, there's a 1 in 2 chance that both are boys. The first child is known to be a boy, and there is a 1 in 2 chance that the next will be a boy.[/hide]

[hide]For the second, there's a 1 in 3 chance that both are boys. There are 4 possibilities for 2 children, each equally likely, but the possibility that both are girls is eliminated. Thus there remain 3 possibilities equally likely, and only 1 of them has both children as boys.[/hide]

[hide]For the third, there's a 13 in 27 chance that both are boys. The trick is to be sure not to double-count the possibility of both boys being born on a Tuesday.[/hide]

 

Person 1 has a .5 chance that both his children are boys. The possible pairs for two children are (Please note the B=Boy, G=Girl, The left one represents the older child):

GB

GG

BG

BB

 

Since we know the first one is a boy then you reduce the number of possibilities to:

 

BG

BB

 

Which means that in 1 out of the 2 scenarios Person 1 has all boys

 

 

 

Person 2 has a 1/3 chance of having all boys. Again the possible pairs for the sex of two children are:

GB

GG

BG

BB

 

Since we know at least one of the children is a boy you can eliminate the possibility that both children are girls. This then leaves you with

GB

BG

BB

 

So out of the three options only one occurs were Person 2 has all boys

 

 

 

For the third person there is a 13/27 chance both are boys. To see why look at the attached PDF where I list all the possible permutations, and then highlighted the ones which give you two boys.

 

 

Warning do not look at the file if you do not want a spoiler to number 3.Born on Tuesday.pdf

Edited July 31, 201015 yr by DJBruce