Question

While hanging Christmas lights for neighbors, Terrell counted the number of broken lights on each string Number of broken lights Number of strings 10 3 42 4 93 6 101 4 135 3 < is the number of broken lights that a randomly chosen string had. What is the expected value of X? Vrite your answer as a decimal.

Answer 1

The expected value is defined by

$E=\Sigma x\cdot P(x)$

This means we have to find the probability of each number of broken lights:

$\begin{gathered} P_1=\frac{3}{20} \\ P_2=\frac{4}{20}=\frac{1}{5} \\ P_3=\frac{6}{20}=\frac{3}{10} \\ P_4=\frac{4}{20}=\frac{1}{5} \\ P_5=\frac{3}{20} \end{gathered}$

Then, we multiply each probability by its x-value, and we add them to find the expected value:

$\begin{gathered} E=10\cdot\frac{3}{20}+42\cdot\frac{1}{5}+93\cdot\frac{3}{10}+101\cdot\frac{1}{5}+135\cdot\frac{3}{20} \\ E=1.5+8.4+27.9+20.2+20.25 \\ E=78.25 \end{gathered}$

Hence, the expected value is 78.25.