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:There is a 28.57% probability that a randomly selected participant who died by the end of the study was a non-smoker.Step-by-step 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[tex]P = \frac{P(B).P(A/B)}{P(A)}[/tex]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[tex]P(B) = 0.6[/tex]P(A/B) is the probability that the participant dies, given that he is a non smoker. So:[tex]P(A/B) = x[/tex]P(A) is the probability that the participant dies:[tex]P(A) = P_{1} + P_{2} + P_{3}[/tex][tex]P_{1}[/tex] is the probability that a heavy smoker is selected and that he dies. So:[tex]P_{1} = 0.15*5x = 0.75x[/tex][tex]P_{2}[/tex] is the probability that a light smoker is selected and that he dies. So:[tex]P_{2} = 0.25*3x = 0.75x[/tex][tex]P_{3}[/tex] is the probability that a non-smoker is selected and that he dies. So:[tex]P_{3} = 0.60*x = 0.60x[/tex]The probability that a participant dies is:[tex]P(A) = P_{1} + P_{2} + P_{3} = 0.75x + 0.75x + 0.60x = 2.10x[/tex]The probability of the participant being a non-smoker, given that he died, is:[tex]P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.6x}{2.10x} = \frac{0.6}{2.10} = 0.2857[/tex]There is a 28.57% probability that a randomly selected participant who died by the end of the study was a non-smoker.