1. 600 deaths were studied and were categorized into the following contingency table:
$\begin{array}{lcc} & \text { Cancer } & \text { Heart Disease } \\ \text { Smoker } & 135 & 310 \\ \text { Non Smoker } & 55 & 100\end{array}$
If one death is randomly selected, find the probability:
a. $P($ cancer $)$
b. $\mathbf{P}($ smoker and heart disease)
c. $P$ (non smoker or heart disease)
d. Find the probability of selecting a non-smoker given that their cause of death was heart disease.
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Answer
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, \(P(\text{non smoker | heart disease}) = \frac{100}{410} = \boxed{0.2439}\).
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, \(P(\text{non smoker | heart disease}) = \frac{100}{410} = \boxed{0.2439}\).
