Question

The number of automobiles sold weekly at a certain dealership is a random variable with an expected value of$16$. Give an upper bound to the probability that

$\left(a\right)$next week’s sales exceed$18$;

$\left(b\right)$next week’s sales exceed$25$.

Short answer

$$\begin{gathered} \left(a\right)P\{X>18\}\le \frac{E\left[X\right]}{18}=\frac{8}{9}=.89 \\ \left(b\right)P\{X>25\}\le \frac{E\left[X\right]}{25}=\frac{16}{25}=.64 \end{gathered}$$

Step-by-step solution

Given Information.

The number of automobiles sold weekly at a certain dealership is a random variable with an expected value of$16$.

Part (a) Explanation.

Let the random variable$X$represent the number of automobiles sold (in a week). It is given that$E\left[X\right]=16$.

By Markov's inequality, we get:

$P\{X>18\}\le \frac{E\left[X\right]}{18}=\frac{16}{18}=\frac{8}{9}=.89$

Part (b) Explanation.

$P\{X>25\}\le \frac{E\left[X\right]}{25}=\frac{16}{25}=\frac{16}{25}=.64$