Orange M&M’S

a. Find and interpret the expected value of $X$.

b. Find and interpret the standard deviation of $X$.

The number of trials, $n=8$

The probability of success, $\begin{array}{rcl}p& =& 20.5\%\\ & =& 0.205\end{array}$

A binomial distribution governs the number of successes among a specified number of independent trials.

Then

The product of sample size $\left(n\right)$and probability $\left(p\right)$determines the expected mean (or value) of a binomial distribution.

$\begin{array}{rcl}\mu & =& np\\ & =& 8\times 0.205\\ & =& 1.64\end{array}$

Thus,

Orange candies make up an average of $1.64$ out of the eight sweets chosen at random.

The Number of trials, $n=8$

The Probability of success, $\begin{array}{rcl}p& =& 20.5\%\\ & =& 0.205\end{array}$

The square root of the product of the sample size $\left(n\right)$and the probabilities $\left(p\right)$and$\left(q\right)$ is the standard deviation of a binomial distribution.

$\begin{array}{rcl}\sigma & =& \sqrt{npq}\\ & =& \sqrt{np(1-p)}\\ & =& \sqrt{8(0.205)(1-0.205)}\\ & =& \sqrt{1.3038}\\ & \approx & 1.1418\end{array}$

Thus,

The amount of orange candies differs by about $1.1418$ candies on average from the mean of $1.64$ candies among 8 randomly picked sweets.