Question

The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. find the probability that fewer than 7 field mice are found

Answer 1

This requires the Poisson distribution, where
area = 5-acres
and mean number of field mice = 12 (in 5-acres of field)
therefore 
lambda=12 (mean, given)
and the probability of k mice in the 5-acre field is given by the Poisson distribution as
P(X=k)=lambda^k * e^(-lambda) / k!   ..............(1)
To find the probability of having LESS than 7 field mice, we add the probabilities of 0 to 6, which is
P(X<7)=P(X=0)+P(X=1)+...+P(X=6)
evaluating with equation (1) for X=0 to 6, we get:
0 0.0000061 0.0000742 0.0004423 0.0017704 0.0053095 0.0127416 0.025481Total = 0.045822

Answer: The probability that fewer than 7 field mice are found in the 5-acre field is 0.0458.