Died from Cancer Did Not Die from Cancer
Never smoked cigars 620 123,094
Former cigar smoker 54 7,161
Current cigar smoker 193 5,839
(a) If an individual is randomly selected from this study, what is the probability that he died from cancer?
(b) If an individual is randomly selected from this study, what is the probability that he was a current cigar smoker?
(c) If an individual is randomly selected from this study, what is the probability that he died from cancer and was a current cigar smoker?
(d) If an individual is randomly selected from this study, what is the probability that he died from cancer or was a current cigar smoker?
The correct answer and explanation is :
To address the given questions, we first need to understand the data provided:
- Never smoked cigars: 620 individuals died from cancer; 123,094 did not.
- Former cigar smoker: 54 individuals died from cancer; 7,161 did not.
- Current cigar smoker: 193 individuals died from cancer; 5,839 did not.
(a) Probability that a randomly selected individual died from cancer:
To find this probability, we need to determine the total number of individuals who died from cancer and divide it by the total number of individuals in the study.
- Total who died from cancer: 620 + 54 + 193 = 867
- Total individuals: (620 + 123,094) + (54 + 7,161) + (193 + 5,839) = 130,961
Thus, the probability is:
[ P(\text{Died from cancer}) = \frac{867}{130,961} \approx 0.0066 ]
(b) Probability that a randomly selected individual was a current cigar smoker:
To find this probability:
- Total current cigar smokers: 193 + 5,839 = 6,032
- Total individuals: 130,961
Thus, the probability is:
[ P(\text{Current cigar smoker}) = \frac{6,032}{130,961} \approx 0.046 ]
(c) Probability that a randomly selected individual died from cancer and was a current cigar smoker:
This is the joint probability of both events occurring:
- Individuals who died from cancer and were current cigar smokers: 193
- Total individuals: 130,961
Thus, the probability is:
[ P(\text{Died from cancer and current cigar smoker}) = \frac{193}{130,961} \approx 0.0015 ]
(d) Probability that a randomly selected individual died from cancer or was a current cigar smoker:
This is the probability of the union of the two events. Using the formula:
[ P(A \cup B) = P(A) + P(B) – P(A \cap B) ]
We already have:
- ( P(A) = P(\text{Died from cancer}) \approx 0.0066 )
- ( P(B) = P(\text{Current cigar smoker}) \approx 0.046 )
- ( P(A \cap B) = P(\text{Died from cancer and current cigar smoker}) \approx 0.0015 )
Thus:
[ P(A \cup B) = 0.0066 + 0.046 – 0.0015 = 0.0511 ]
Explanation:
These probabilities provide insights into the relationships between smoking habits and cancer mortality within the study population. Notably, while current cigar smokers constitute approximately 4.6% of the population, they represent a higher proportion of cancer deaths (22.2%) compared to their prevalence. This suggests a significant association between current cigar smoking and increased cancer mortality risk. Additionally, the combined probability of either dying from cancer or being a current cigar smoker is about 5.11%, highlighting the impact of smoking on cancer-related deaths.