Question

Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and eight who are not. We are interested in the number on the committee who are not technically proficient.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many instructors do you expect on the committee who are not technically proficient?

e. Find the probability that at least five on the committee are not technically proficient.

f. Find the probability that at most three on the committee are not technically proficient.

Short answer

a. The random variable X is the number of people on the committee who are not technically proficient.

b. The values of X are $X=0,1,2,3,4,5,6,7,8,9,10$

c. The distribution of X is $X~B(8,20,10)$

d. The expected number of instructors on the committee who are not technically proficient is $2.8571$

e. The probability that at least five on the committee are not technically proficient$0.7721$

f. The probability that at most three on the committee are not technically proficient is$0.71604$

Step-by-step solution

Content Introduction

A discrete probability distribution, the hypergeometric distribution. It's utilized when you want to know how likely it is to get a certain number of successes from a given sample size without replacement

Explanation (part a)

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case X is the number of people on the committee who are not technically proficient.

Explanation (part b)

Make the list of values that you want to use X may take on.

As we can see there is an upper bound for the situation at hand so,

$X=0,1,2,3,4,5,6,7,8,9,10$

Explanation (part c)

The random variable X refers to the number of trials before the first success. Each trial is independent of others and has similar probability of success.

The distribution of X is$X~B(8,20,10)$

Explanation (part d)

The expected number of instructors on the committee who are not technically proficient is :

$E\left(X\right)=\frac{n\times r}{r+b}$

where, $n=10,r=8,b=20$

$$\begin{gathered} E\left(X\right)=\frac{10\times 8}{8+20} \\ E\left(X\right)=2.8571 \end{gathered}$$

Explanation (part e)

To find the probability of at least five means it can be five or less than five.

The probability can also be shown as,

$\begin{array}{r}P(x\ge 5)=1-P(x<5)\\ P(x\ge 5)=1-P(x\le 4)\end{array}$

The probability that at least five on the committee are not technically proficient is:

$$\begin{gathered} P(X\ge 5)=1-P(X\le 4) \\ P(X\ge 5)=1-0.92279 \\ P(X\ge 5)=0.07721 \end{gathered}$$

Explanation (part f)

To find the probability of at most three, that means the probability should be more than three.

The probability that at most three on the committee are not technically proficient is:

Using an excel sheet we can find the the required probability,

$P(X\le 3)=0.71604$