Question

(a) Explain why$X$is a binomial random variable.

(b) Find and interpret the expected value of$X$.

(c) Find and interpret the standard deviation of$X$.

Short answer

(a) Variable$x$has all the required conditions for a binomial distribution Thus, the variable $x$is a binomial random variable.

(b) Expected value of is $1.60$

(c) Standard deviation of $X$is$1.1314$

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that, According to the Mars candy company, $20\%$of its plain M&M’s candies are orange. Assume that the company’s claim is true. Suppose that you reach into a large bag of plain M&M’s (without looking) and pull out $8$candies.

Part (a) Step 2: Explanation

Probability of success $(p)$localid="1649747575227" $=20\%=0.20$.

Number of trials localid="1649747578779" $\left(n\right)=8$

The random variable $X$, which is a binomial random variable, is as follows:

• There are two types of successes and failures: orange candy and non-orange candy.
• The candies are kept apart from one another.
• The number of candies is predetermined.
• The likelihood of receiving orange candy is set.

Because all of the requirements for a binomial distribution are predetermined. As a result, the variable $X$is a random variable with a binomial distribution.

Part (b) Step 1: Given information

Given in the question that, According to the Mars candy company, $20\%$of its plain M&M’s candies are orange. Assume that the company’s claim is true. Suppose that you reach into a large bag of plain M&M’s (without looking) and pull out $8$
candies. Let $X$= the number of orange candies you get. We need to find and interpret the expected value of $X$.

Part (b) Step 2: Explanation

The mean can be calculated as:

$\mu =n\times p$

$=8(0.20)$

$=1.60$

The above computed mean shows that the expected number of candies is 1 .

Part (c) Step 1: Given information

Given in the question that, According to the Mars candy company, $20\%$of its plain M&M’s candies are orange. Assume that the company’s claim is true. Suppose that you reach into a large bag of plain M&M’s (without looking) and pull out $8$candies. Let $X$= the number of orange candies you get. We need to find and interpret the standard deviation of $X$.

Part (c) Step 2: Explanation

The standard deviation can be calculated as:

$\sigma =\sqrt{n\times p\times (1-p)}$

$=\sqrt{8\left(0.20\right)(1-0.20)}$

$=1.1314$

The above-computed mean indicates that the number of candies will vary by $1.1314$by the average of $1.60$candies.