Chapter 5.3, Problem 78E

To determine

Probability for the randomly selected adult belongs to a health club and goes there at least twice a week.

Answer to Problem 78E

Probability for the randomly selected adult belongs to a health club and goes there at least twice a week is 0.04.

Explanation of Solution

Given information:

10% of the adults belong to health clubs.

40% of these health club members go to the club at least twice a week.

Calculations:

According to general multiplication rule,

  $P\left(\text{A}\text{and}\text{B}\right)=P\left(\text{A}\cap \text{B}\right)=P\left(\text{A}\right)\times P\left(\text{B}|\text{A}\right)=P\left(\text{B}\right)\times P\left(\text{A}|\text{B}\right)$

Let

H: adults belong to health clubs

W: health club members go to the club at least twice a week

Now,

The corresponding probabilities:

Probability for the adults belong to health clubs,

  $P\left(\text{H}\right)=0.10$

Probability for the health club members go to the club at least twice a week,

  $P\left(\text{W}|\text{H}\right)=0.40$

Apply the general multiplication rule:

  $\begin{array}{l}P\left(\text{H}\text{and}\text{W}\right)=P\left(\text{H}\cap \text{W}\right)=P\left(\text{H}\right)\times P\left(\text{W}|\text{H}\right)\\ =0.10\times 0.40\\ =0.04\end{array}$

Thus,

Probability for a randomly selected adult belongs to a health club and visit club at least twice a week is 0.04.