The milkweed aphid is a common pest to many ornamental plant

The milkweed aphid is a common pest to many ornamental plants. Suppose that the number of aphids on a shoot of a Mexican butterfly weed follows a Poisson distribution with =5 aphids.

a) What is the probability (±0.0001) of observing exactly 6 aphids on a shoot?

b) What is the probability (±0.0001) of observing 6 or fewer aphids on a shoot?

c) What proportion (±0.0001) of shoots would you expect to have no aphids present?

Solution

a) P(X = 6) = e^-5 * 5⁶ / 6! = 0.1462

b) P(X < 6) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)

                  = e^-5 * 5⁰ / 0! + e^-5 * 5¹ / 1! + e^-5 * 5² / 2! + e^-5 * 5³ / 3! + e^-5 * 5⁴ / 4! + e^-5 * 5⁵ / 5! + e^-5 * 5⁶ / 6!

                  = 0.6160

c) P(X = 0) = e^-5 * 5⁰ / 0! = 0.0067