Question

At the beginning of a study of individuals, \(15 \%\) were classified as heavy smokers, \(30 \%\) were classified as light smokers, and \(55 \%\) were classified as nonsmokers. In the five-year study, it was determined that the death rates of the heavy and light smokers were five and three times that of the nonsmokers, respectively. A randomly selected participant died over the five- year period: calculate the probability that the participant was a nonsmoker.

Short answer

The probability that a participant who died was a nonsmoker is \(0.55 * x\) given the assumption that \(P(D) = 1\) for the purpose of this calculation.

Step-by-step solution

Compute the probabilities of dying

Firstly, assign the given probabilities to variables: \(P(H) = 0.15\) for heavy smokers, \(P(L) = 0.30\) for light smokers, and \(P(N) = 0.55\) for nonsmokers. Given that death rates for heavy and light smokers are five and three times that of the nonsmokers respectively, let's assume the probability of a nonsmoker dying (in the absence of any given value) is \(x\). Therefore, the probability of a heavy smoker dying is \(5x\), and for a light smoker it is \(3x\).

Compute the total probability of death

The total probability of dying is given by the sum of the probabilities of dying for each group, weighted by the size of each group: \(P(D) = P(H) * 5x + P(L) * 3x + P(N) * x = 0.15 * 5x + 0.30 * 3x + 0.55 * x\). However, as \(x\) is unknown, assume the total probability of death, \(P(D)\), equals to 1 for the purpose of this calculation.

Calculate the conditional probability

The probability that a participant who died was a nonsmoker is written as \(P(N|D)\) and given by \(P(N \cap D) / P(D)\). Given the probabilities calculated in Step 1, \(P(N \cap D) = P(N) * x = 0.55 * x\). So, \(P(N|D) = P(N \cap D) / P(D) = (0.55 * x) / 1 = 0.55 * x\).

Interpret the result

The quantity \(0.55x\) represents the chance that a nonsmoker will die over the five-year period. When expressed relative to the total probability of death, it gives the conditional probability that someone who died over this period was a nonsmoker.