in a certain town 40% of men are smokers. It is known that 6% of men in this town have lung disease. Of those men having lung disease 85% are smokers. A man is selected at random from this town. Consider the events: D<<the selected man has a lung disease>> and S<<the selected man is a smoker>>.
1)Calculate the probability of<<the selected man is a smoker and has lung disease>>
2)Calculate the probability of << the selected man is a nonsmoker and has a lung disease>>
3)The selected man does not have a lung disease. Calculate the probability that he is a smoker.
4)Calculate the probability of<<the selected man has a lung disease knowing that he is a smoker.
To calculate the probabilities, we can use the concept of conditional probability. Here's how you can approach each question:
1) To find the probability of the selected man being a smoker and having lung disease, we need to multiply the probability of being a smoker (40%) by the probability of having lung disease given that the man is a smoker (85%):
P(S and D) = P(S) * P(D|S) = 0.40 * 0.85 = 0.34
So, the probability of the selected man being a smoker and having lung disease is 0.34.
2) To find the probability of the selected man being a non-smoker and having lung disease, we need to multiply the probability of being a non-smoker (1 - 0.40 = 0.60) by the probability of having lung disease given that the man is a non-smoker (100% - 85% = 15%):
P(~S and D) = P(~S) * P(D|~S) = 0.60 * 0.15 = 0.09
So, the probability of the selected man being a non-smoker and having lung disease is 0.09.
3) To find the probability that the selected man, who does not have a lung disease, is a smoker, we need to use Bayes' theorem. We want to calculate P(S|~D), which is the probability of being a smoker given that the man does not have lung disease.
P(S|~D) = (P(~D|S) * P(S)) / P(~D)
P(~D|S) is the probability of not having lung disease given that the man is a smoker, which is calculated as (100% - 85% = 15%).
P(~D) is the probability of not having lung disease in the general population, which can be found by subtracting the probability of having lung disease (6%) from 100%.
P(S|~D) = (0.15 * 0.40) / (1 - 0.06) = 0.06 / 0.94 ≈ 0.064
So, the probability of the selected man, who does not have lung disease, being a smoker is approximately 0.064.
4) To find the probability of a selected man having lung disease given that he is a smoker, we use Bayes' theorem again. We want to calculate P(D|S), which is the probability of having lung disease given that the man is a smoker.
P(D|S) = (P(S|D) * P(D)) / P(S)
P(S|D) is the probability of being a smoker given that the man has lung disease, which is given as 85%.
P(D) is the probability of having lung disease in the general population, which can be found directly as 6%.
P(S) is the probability of being a smoker in the general population, which is given as 40%.
P(D|S) = (0.85 * 0.06) / 0.40 = 0.051 / 0.40 ≈ 0.128
So, the probability of the selected man, who is a smoker, having lung disease is approximately 0.128.