if square i has exactly one badger set in it,
for i = 1, 2, 3, ... , 36.
Assuming the badger sets make their choices independently and that the squares are all equally likely to be chosen, the number of sets in a square follows a Binomial distribution with n = 42 and p = 1/36. So
This is also .
The expected value of the number of squares with exactly one badger set is then
, which is approximately 13.23.
We have used the theorem that E(X+Y) = E(X) + E(Y) above. It's important to realize that the theorem does not require that X and Y be independent. That is good for us, because the 's are not independent.