Question

According to a recent Pew Research poll, 75% of millenials (people born between 1981 and 1995) have a profile on a social networking site. Let X = the number of millenials you ask until you find a person without a profile on a social networking site.

a. Describe the distribution of X.

b. Find the (i) mean and (ii) standard deviation of X.

c. What is the probability that you must ask ten people to find one person without a social networking site?

d. What is the probability that you must ask 20 people to find one person without a social networking site?

e. What is the probability that you must ask at most five people?

Short answer

a. The distribution of X is $X~G(0.25)$

b. The mean is $4$and standard deviation is $3.4641$

c. The probability that ten people find one person without a social networking site is $0.0188$

d. The probability that $20$people to find one person without a social networking site is $0.0011$

e. The probability that at most five people$0.7627$

Step-by-step solution

Content Introduction

In a Bernoulli trial, the likelihood of the number of successive failures before a success is obtained is represented by a geometric distribution, which is a sort of discrete probability distribution.

Explanation (part a)

We are given, $75\%or0.75$of millennials who have a profile on a social networking site and in order to find a person without a profile on a social networking site is$(1-0.75=0.25)$.

Make the list of values that you want the X may take on where an upper bound for the situation can be seen at hand so,

$X~G(0.25)$

Explanation (part b)

The mean in geometric distribution is $mean=\frac{1}{p}$where $p=0.25$as we are to find the person without a profile on a social networking site

$$\begin{gathered} mean=\frac{1}{p} \\ mean=\frac{1}{0.25} \\ mean=4 \end{gathered}$$

Standard deviation is $\sqrt{\frac{1-p}{p^2}}$where $p=0.25$

$$\begin{gathered} \delta =\sqrt{\frac{1-p}{p^2}} \\ \delta =\sqrt{\frac{1-0.25}{{(0.25)}^2}} \\ \delta =3.4641 \end{gathered}$$

Explanation (part c)

The probability that ten people find one person without a social networking site is calculated as follow:

$$\begin{gathered} P(X=10)={(1-0.25)}^{x-1}\times (0.25) \\ P(X=10)={(0.75)}^9(0.25) \\ P(X=10)=0.0188 \end{gathered}$$

Explanation (part d)

The probability that $20$people find one person without a social networking site is calculated as follow:

role="math" localid="1649166303751" $$\begin{gathered} P(X=20)={(1-0.25)}^{x-1}(0.25) \\ P(X=20)={(0.75)}^{19}(0.25) \\ P(X=20)=0.0011 \end{gathered}$$

Explanation (part e)

The probability that at most five people is calculated as follow:

$$\begin{gathered} P(X\le 5)=P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5) \\ P(X\le 5)=0.7627 \end{gathered}$$