austin works at an ice cream shop and decided to record some data about the number of scoops of ice cream…

Explanation:

Step1: Recall expected - value formula

The formula for the expected value $E(X)$ of a discrete - random variable is $E(X)=\sum_{i}x_ip_i$, where $x_i$ are the possible values and $p_i$ are their corresponding probabilities.

Step2: Identify values and probabilities

Here, $x_1 = 1$, $p_1=\frac{1}{3}$; $x_2 = 2$, $p_2=\frac{1}{6}$; $x_3 = 3$, $p_3=\frac{1}{2}$.

Step3: Calculate the product for each pair

For the first pair: $x_1p_1=1\times\frac{1}{3}=\frac{1}{3}$. For the second pair: $x_2p_2=2\times\frac{1}{6}=\frac{1}{3}$. For the third pair: $x_3p_3=3\times\frac{1}{2}=\frac{3}{2}$.

Step4: Sum up the products

$E(X)=\frac{1}{3}+\frac{1}{3}+\frac{3}{2}=\frac{2 + 2+9}{6}=\frac{13}{6}\approx2.17$.

Answer:

$\frac{13}{6}$