The number of combines sold each year by a dealership that specialises in agricultural equipment is a Poisson random variable with an average of 4.
What is the probability that the dealership will sell:
3.1Less than five combines in a given year.
To find the probability that the dealership will sell less than five combines in a given year, we can use the Poisson distribution formula:
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Where X is the number of combines sold.
Using the Poisson distribution formula with an average of 4 combines sold per year:
P(X = k) = (e^(-4) * 4^k) / k!
Calculating each individual probability and summing them up:
P(X = 0) = (e^(-4) * 4^0) / 0! = e^(-4) ≈ 0.0183
P(X = 1) = (e^(-4) * 4^1) / 1! = 4 * e^(-4) ≈ 0.0733
P(X = 2) = (e^(-4) * 4^2) / 2! = 8 * e^(-4) ≈ 0.1465
P(X = 3) = (e^(-4) * 4^3) / 3! = 32 * e^(-4) ≈ 0.1953
P(X = 4) = (e^(-4) * 4^4) / 4! = 64 * e^(-4) / 24 ≈ 0.1953
Adding them up:
P(X < 5) = 0.0183 + 0.0733 + 0.1465 + 0.1953 + 0.1953 ≈ 0.6287
Therefore, the probability that the dealership will sell less than five combines in a given year is approximately 0.6287.