Question

smokers are interested in the relationship between cigarette smoking and lung cancer. suppose an adult is randomly selected from a particular population. the following table shows the probabilities of some events related to this adults profile.

(a) consider a venn diagram for representing the outcomes of this chance process. complete the venn diagram by dragging the values in the answer box into the venn diagram.

(b) find the probability that the individual gets cancer given that they are a smoker.

(round to 2 decimal places. leave your answer in decimal form.)

(c) find the probability that the individual smokes or gets cancer.

(round to 2 decimal places. leave your answer in decimal form.)

(d) two adults are selected at random. find the probability that at least one of the two smokes.

(round to 4 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Probability of cancer given smoker (part b)

Use the formula for conditional probability \(P(A|B)=\frac{P(A\cap B)}{P(B)}\). Here, \(A\) is having cancer and \(B\) is being a smoker. \(P(\text{smoker}\cap\text{cancer}) = 0.08\), \(P(\text{smoker})=0.25\). So \(P(\text{cancer}|\text{smoker})=\frac{0.08}{0.25}=0.32\)

Step2: Probability of smoker or cancer (part c)

Use the formula \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). Let \(A\) be smoker (\(P(A) = 0.25\)), \(B\) be cancer (\(P(B)=0.08 + 0.09=0.17\)), \(P(A\cap B)=0.08\). Then \(P(A\cup B)=0.25+0.17 - 0.08=0.34\)

Step3: Probability of at least one smoker (part d)

First find the probability that neither is a smoker. The probability of not being a smoker is \(1 - 0.25=0.75\). For two independent individuals, the probability that neither smokes is \(0.75\times0.75 = 0.5625\). Then the probability that at least one smokes is \(1-0.5625 = 0.4375\)

Answer:

(b) \(0.32\)
(c) \(0.34\)
(d) \(0.4375\)