Question

a health study tracked a group of persons for five years. at the beginning of the study 21% were classified as heavy smokers, 35% as light smokers, and 44% as nonsmokers. results of the study showed that light smokers were twice as likely as nonsmokers to die during the five - year study but only half as likely as heavy smokers. a randomly selected participant from the study dies over the five - year period. calculate the probability that the participant was a heavy smoker. (hint: let x = p(a nonsmoker dies).) which of the following is the probability, rounded to two decimal places? a. 0.42 b. 0.33 c. 0.27 d. 0.55 e. 0.23

Explanation:

Step1: Define probabilities

Let $P(\text{nonsmoker}) = 0.44$, $P(\text{light - smoker})=0.35$, $P(\text{heavy - smoker}) = 0.21$. Let $x = P(\text{a nonsmoker dies})$. Then $P(\text{a light - smoker dies}) = 2x$ and $P(\text{a heavy - smoker dies})=4x$.

Step2: Use the law of total probability

The probability that a randomly - selected person dies, $P(D)$ is given by:
\[

$$\begin{align*} P(D)&=P(\text{nonsmoker})\times P(\text{dies}|\text{nonsmoker})+P(\text{light - smoker})\times P(\text{dies}|\text{light - smoker})+P(\text{heavy - smoker})\times P(\text{dies}|\text{heavy - smoker})\\ &=0.44x + 0.35\times2x+0.21\times4x\\ &=0.44x + 0.7x+0.84x\\ &=(0.44 + 0.7+0.84)x\\ &=1.98x \end{align*}$$

\]

Step3: Use Bayes' theorem

We want to find $P(\text{heavy - smoker}|D)$. By Bayes' theorem, $P(\text{heavy - smoker}|D)=\frac{P(\text{heavy - smoker})\times P(\text{dies}|\text{heavy - smoker})}{P(D)}$.
Substitute the values we found: $P(\text{heavy - smoker})\times P(\text{dies}|\text{heavy - smoker})=0.21\times4x = 0.84x$ and $P(D)=1.98x$.
So $P(\text{heavy - smoker}|D)=\frac{0.84x}{1.98x}=\frac{0.84}{1.98}\approx0.42$.

Answer:

A. 0.42