The gestation period for a particular breed of cow is normal

The gestation period for a particular breed of cow is normally distributed with a mean of 280 days and a standard deviation of 3 days. A rancher who runs a cow-calf business keeps one bull in the field with the cows at all times. He sells an old bull and replaces it with a young one. A cow has a calf 288 days later. The farmer asks you which bull sired the calf. Use a probabilistic argument to respond to him.

Solution

p(z<288)=288-280/3^2 = 0.89 p(z>288)=1-0.89=0.11 so there is 0.89 probability that the young bull sired the cow and 0.11 probability that the old bull sired the cow