# July 2016 Brain Teaser Solution

Q: In the town of Mathville, Anne is 5 years old.
The composition of the families in Mathville are as follows:
2% include five or more children
7% include four children
14% include three children
31% include two children
16% include one child
30% do not include any children

What are the chances that Anne lives with two brothers and no sisters?
(Consider all children living in any given house as siblings)

First we have to assume that there is a 50-50 chance for each child’s gender (although studies have shown if you already have 2 boys, you are likelier to have another son rather than a daughter).
We also have to assume that 5 is the number of children of the very large families.

Since Anne is a child, she cannot possibly live in a household with no children.
There is a (3×14)/(0 x 30 + 1 x 16 + 2 x 31 + 3 x 14 + 4 x 7 + 5 x 2)
or 26.582% chance that she lives with two other children.   Now we have to figure out the probability that both of the other children are boys.

There is a 50% chance that each of these other children is a boy, so there is a 25% chance that they are both boys (the combinations for the two other children are: BB, BG, GB and GG),   Therefore, the probability that Anne lives with two brothers is 25% of 26.582%, or about 6.6%.