Step 1 :Given a contingency table with the number of deaths due to cancer and heart disease for smokers and non-smokers, we can calculate the probability of each event by dividing the number of occurrences of the event by the total number of deaths.
Step 2 :For part a, we need to find the probability of a death being due to cancer. This can be calculated by adding the number of smokers and non-smokers who died of cancer and dividing by the total number of deaths. So, \(P(\text{cancer}) = \frac{190}{600} = \boxed{0.3167}\).
Step 3 :For part b, we need to find the probability of a death being due to heart disease and the person being a smoker. This can be calculated by dividing the number of smokers who died of heart disease by the total number of deaths. So, \(P(\text{smoker and heart disease}) = \frac{310}{600} = \boxed{0.5167}\).
Step 4 :For part c, we need to find the probability of a death being due to heart disease or the person being a non-smoker. This can be calculated by adding the number of non-smokers and the number of deaths due to heart disease and dividing by the total number of deaths. However, we need to subtract the number of non-smokers who died of heart disease because we are counting them twice. So, \(P(\text{non smoker or heart disease}) = \frac{365}{600} = \boxed{0.6083}\).
Step 5 :For part d, we need to find the probability of a person being a non-smoker given that their cause of death was heart disease. This can be calculated by dividing the number of non-smokers who died of heart disease by the total number of deaths due to heart disease. So, \(P(\text{non smoker | heart disease}) = \frac{100}{410} = \boxed{0.2439}\).