Question

In Exercises 25–28, find the probabilities and answer the questions.

Too Young to Tat Based on a Harris poll, among adults who regret getting tattoos, 20% say that they were too young when they got their tattoos. Assume that five adults who regret getting tattoos are randomly selected, and find the indicated probability.

a. Find the probability that none of the selected adults say that they were too young to get tattoos.

b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos.

c. Find the probability that the number of selected adults saying they were too young is 0 or 1.

d. If we randomly select five adults, is 1 a significantly low number who say that they were too young to get tattoos?

Short answer

a. The probability that none of the adults says they were too young to get tattoos is equal to 0.32768.

b. The probability that exactly 1 adult says that he/she was too young to get tattoos is equal to 0.4096.

c.The probability that the number of adults saying they were too young is 0 or 1 isequal to 0.73728.

d. It can be said that out of 5 adults, 1 is not a significantly low number who say that they were too young to get tattoos.

Step-by-step solution

Given information

It is given that 20% of people say that they were too young when they got their tattoos.

A sample of 5adults is selected.

Required probabilities

Let X denote the number of adults who say they were too young when they got their tattoos.

Let success be defined as getting an adult who says that he/she was too young when they got their tattoos.

The number of trials (n) is given to be equal to 5.

The probability of success is given as follows:

$$\begin{gathered} p=20\% \\ =\frac{20}{100} \\ =0.20 \end{gathered}$$

The probability of failure is given as follows:

$$\begin{gathered} q=1-p \\ =1-0.20 \\ =0.80 \end{gathered}$$

The following binomial probability formula is used:

$P\left(X=x\right)={}^n{C}_x{\left(p\right)}^x{\left(q\right)}^{n-x}$

a.

The number of successes required in 5 trials should be x=0.

Using the binomial probability formula, the probability that none of the adults says that they were too young to get tattoos is computed below:

$$\begin{gathered} P\left(X=0\right)={}^5{C}_0{\left(0.20\right)}^0{\left(0.80\right)}^{5-0} \\ =0.32768 \end{gathered}$$

Therefore, the probability that none of the adults says they were too young to get tattoos is equal to 0.32768.

b.

The number of successes required in 5 trials should be x=1.

Using the binomial probability formula, the probability that exactly 1 adult says that he/she was too young to get tattoos is computed below:

$$\begin{gathered} P\left(X=1\right)={}^5{C}_1{\left(0.20\right)}^1{\left(0.80\right)}^{5-1} \\ =0.4096 \end{gathered}$$

Therefore, the probability that exactly 1 adult says he/she was too young to get tattoos is equal to 0.4096.

c.

The probability that the number of adults saying they were too young is 0 or 1 is computed below:

$$\begin{gathered} P\left(X⩽1\right)=P\left(X=0\right)+P\left(X=1\right) \\ ={}^5{C}_0{\left(0.20\right)}^0{\left(0.80\right)}^{5-0}+{}^5{C}_1{\left(0.20\right)}^1{\left(0.80\right)}^{5-1} \\ =0.73728 \end{gathered}$$

Therefore, the probability that the number of adults saying they were too young is 0 or 1 is equal to 0.73728.

Significance of the probability

d.

The number of successes (x) of a binomial probability value is said to be significantly low if

$P\left(x\text{or}\text{fewer}\right)⩽0.05$.

Here, the number of successes (people who believe they were too young to get tattoos) is equal to 1.

$$\begin{gathered} P\left(1\text{or}\text{fewer}\right)=0.73728 \\ >0.05 \end{gathered}$$

Thus, it can be said that out of 5 adults, 1 is not a significantly low number who say that they were too young to get tattoos.