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

See You Later Based on a Harris Interactive poll, 20% of adults believe in reincarnation. Assume that six adults are randomly selected, and find the indicated probability.

a. What is the probability that exactly five of the selected adults believe in reincarnation?

b. What is the probability that all of the selected adults believe in reincarnation?

c. What is the probability that at least five of the selected adults believe in reincarnation?

d. If six adults are randomly selected, is five a significantly high number who believe in reincarnation?

It is given that 20% of adults believe in reincarnation.

A sample of 6 adults is selected.

Let X denote the number of adults who believe in reincarnation.

Let success be defined as getting an adult who believes in reincarnation.

The number of trials (n) is given to be equal to 6.

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 6 trials should be x=5.

Using the binomial probability formula, the probability that exactly 5 adults believe in reincarnation is computed below:

$$\begin{gathered} P\left(X=5\right)={}^6{C}_5{\left(0.20\right)}^5{\left(0.80\right)}^{6-5} \\ =0.00154 \end{gathered}$$

Therefore, the probability that exactly 5 adults believe in reincarnation is equal to 0.00154.

b.

The number of successes required in 6 trials should be x=6.

Using the binomial probability formula, the probability that all 6 adults believe in reincarnation is computed below:

$$\begin{gathered} P\left(X=6\right)={}^6{C}_6{\left(0.20\right)}^6{\left(0.80\right)}^{6-6} \\ =0.000064 \end{gathered}$$

Therefore, the probability that all 6 adults believe in reincarnation is equal to 0.000064.

c.

The number of successes required in 6 trials should be at least equal to 5.

Using the binomial probability formula, the probability that at least 5 adults believe in reincarnation is computed below:

$$\begin{gathered} P\left(X⩾5\right)=P\left(X=5\right)+P\left(X=6\right) \\ ={}^6{C}_5{\left(0.20\right)}^5{\left(0.80\right)}^{6-5}+{}^6{C}_6{\left(0.20\right)}^6{\left(0.80\right)}^{6-6} \\ =0.0016 \end{gathered}$$

Therefore, the probability that at least 5 adults believe in reincarnation is equal to 0.0016.

d.

The number of successes (x) of a binomial probability value is said to be significantly high if$P\left(x\text{or}\text{more}\right)⩽0.05$.

Here, the number of successes (people who believe in reincarnation) is equal to 5.

$$\begin{gathered} P\left(5\text{or}\text{more}\right)=0.0016 \\ <0.05 \end{gathered}$$

Thus, it can be said that 5 is a significantly high number of adults who believe in reincarnation.