Chapter 6, Problem 23T

Solution

To Calculate:

The probability that out of randomly selected $8$ people at least $2$ will cite fishing as their favorite leisure time activity.

  $0.158$

Given:

A study found that $9\%$ of people cite fishing as their favorite leisure time activity.

Concepts Used:

In a binomial experiment with $n$ independent trials each having probability of success $p$ . The probability of $r$ successes is given by ${}_n{C}_r{p}^r{\left(1-p\right)}^{n-r}$ .

The probability of occurrence of the union of many events that are mutually exclusive is the sum of their individual probabilities.

The definition ${}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}$ .

The probability of occurrence of an event and that of its complement add up to $1$ .

Calculations:

The probability that fishing is cited as the favorite leisure time activity is $9\%$ . The favorite leisure time activity of each person is independent. Calculate the probability that at least $2$ out of randomly selected $8$ people will cite fishing as their favorite leisure time activity. The experiment is equivalent to a binomial experiment with $8$ independent trials each having probability of success $0.09$ .

  $\begin{array}{c}P\left(\text{at least 2 cite fishing}\right)=1-P\left(\text{less than 2 cite fishing}\right)\\ =1-P\left(\text{none cite fishing}\right)-P\left(\text{one cites fishing}\right)\\ =1{-}_8{C}_0{\left(0.09\right)}^0{\left(0.91\right)}^8{-}_8{C}_1{\left(0.09\right)}^1{\left(0.91\right)}^7\\ =1-{\left(0.91\right)}^7\left(0.91+8\times 0.09\right)\\ \approx \text{0}\text{.158}\end{array}$

Conclusion:

  $0.158$ is the probability that at least $2$ out of randomly selected $8$ people will cite fishing as their favorite leisure time activity.