Question

In a certain clinical study, 15% of participants were classified as heavy smokers, 25% as light-smokers, and the rest as non-smokers. At the end of the study, the death rates of the heavy and light smokers were 5 and 3 times that of non-smokers, respectively. What is the probability that a randomly selected participant who died by the end of the study was a non-smoker?

Answer 1

There is a 28.57% probability that a randomly selected participant who died by the end of the study was a non-smoker.

Explanation:

We have the following probabilities:

A 15% probability that a participant is classified as a heavy smoker.

A 25% probability that a participant is classified as a light smoker.

A 100% - 25% - 15% = 60% probability that a participant is classified as a non smoker.

A x% probability that a non smoker dies.

A 3x% probability that a light smoker dies.

A 5x% probability that a heavy smoker dies.

This can be formulated as the following problem:

What is the probability of B happening, knowing that A has happened.

It can be calculated by the following formula

`P = (P(B).P(A/B))/(P(A))`

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

This problem is:

What is the probability of the participant being a non-smoker, given that he died?

P(B) is the probability that the participant is a non smoker. So

`P(B) = 0.6`

P(A/B) is the probability that the participant dies, given that he is a non smoker. So:

`P(A/B) = x`

P(A) is the probability that the participant dies:

`P(A) = P_(1) + P_(2) + P_(3)`

`P_(1)` is the probability that a heavy smoker is selected and that he dies. So:

`P_(1) = 0.15*5x = 0.75x`

`P_(2)` is the probability that a light smoker is selected and that he dies. So:

`P_(2) = 0.25*3x = 0.75x`

`P_(3)` is the probability that a non-smoker is selected and that he dies. So:

`P_(3) = 0.60*x = 0.60x`

The probability that a participant dies is:

`P(A) = P_(1) + P_(2) + P_(3) = 0.75x + 0.75x + 0.60x = 2.10x`

The probability of the participant being a non-smoker, given that he died, is:

`P = (P(B).P(A/B))/(P(A)) = (0.6x)/(2.10x) = (0.6)/(2.10) = 0.2857`

There is a 28.57% probability that a randomly selected participant who died by the end of the study was a non-smoker.