3.19 - A First Course in Probability

Question

A total of 48 percent of the women and 37 percent of the men who took a certain “quit smoking” class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of a year. If 62 percent of the original class was male, 

(a) what percentage of those attending the party were women? 

(b) what percentage of the original class attended the party? 

Step-by-Step Solution

Answer

$\text{a. The percentage of those attending the party were women is }44.29\%\text{. }$

$\text{b. The percentage of the original class attended the party is }41.18\%\text{. }$

Step 1 : Given information

Probability of women that attended the party is 0.38.

So, the probability of women that attended the party is 1-0.38=0.62

The probability that the person attended the party given she is woman is 0.48.

The probability that the person attended the party given she is woman is 0.37

Step 2 : Calculation of solution (Part a)

$\text{ The percentage of women that attended the party can be calculated as: }$

$P(\text{ Women }\mid \text{ Attended the party })=\frac{P\left(\text{ Women and Attended the party }\right)}{P(\text{ Attended the party) }}$

                                                        $=\frac{0.48(0.38)}{0.48(0.38)+(0.37)(0.62)}$

                                                        $=0.4429$

                                                        $=44.29\%$

$\text{ Therefore, the required percentage is }44.29\%\text{. }$

Step 3 : Calculation of solution (Part b)

$\text{ The percentage of original class that attended the party can be calculated as: }$

$P\left(\begin{array}{l}\text{ Attended the }\\ \text{ party }\end{array}\right)=P\left(\begin{array}{l}\text{ Attended the }\\ \text{ party|Women }\end{array}\right)P\left(\text{ Women }\right)+P\left(\begin{array}{c}\text{ Attended the }\\ \text{ party }|Men\end{array}\right)P\left(Men\right)$

                                $=0.48(0.38)+0.37(0.62)$

                                $=0.4118$

                                $=41.18\%$

$\text{ Thus, the required percentage is }41.18\%\text{. }$