Question

The auto parts department of an automotive dealership sends out a mean of 4.2 special orders daily. What is the probability that, for any day, the number of special orders sent out will be exactly 3? Round your answer to four decimal places.

Answer 1

The probability that, for any day, the number of special orders sent out will be exactly 3 is 0.1633.

What are some of the properties of Poisson distribution?

The Poisson distribution is a discrete probability distribution that describes the likelihood of a specific number of events occurring in a specified span of time or space, if these events occur at a constant mean rate and regardless of the time since the last occurrence.

Let X ~ Pois(λ)

Then we have:

E(X) = λ = Var(X)

Since standard deviation is the square root (positive) of variance,

Thus,

The standard deviation of X = $\sqrt{\lambda}$

Its probability function is given by

$f(k; λ) = Pr(X = k) = \dfrac{\lambda^{k}e^{-\lambda}}{k!}$

Given the mean is 4.2.

We have to find the probability that on any day, the number of special orders sent out will be exactly 3. Therefore, the value of x will be 5.

Using the Poisson distribution,

$P(x) = \dfrac{e^{-\lambda}\lambda ^x}{x!}\\\\P(5) =\dfrac{e^{-4.2}(4.2)^5}{5!}\\\\P(5) = 0.1633$

Hence, the probability that, for any day, the number of special orders sent out will be exactly 3 is 0.1633.

Learn more about Poisson distribution here:

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