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 :
Let’s analyze the provided data to answer the given probability questions:
Data Breakdown:
- Never smoked cigars:
- Died from cancer: 620
- Did not die from cancer: 123,094
- Total: 123,714
- Former cigar smoker:
- Died from cancer: 54
- Did not die from cancer: 7,161
- Total: 7,215
- Current cigar smoker:
- Died from cancer: 193
- Did not die from cancer: 5,839
- Total: 6,032
Overall Totals:
- Died from cancer: 620 + 54 + 193 = 867
- Did not die from cancer: 123,094 + 7,161 + 5,839 = 136,094
- Grand total: 867 + 136,094 = 136,961
(a) Probability of Dying from Cancer:
The probability (P) that a randomly selected individual died from cancer is calculated by dividing the number of individuals who died from cancer by the total number of individuals:
[ P(\text{Died from cancer}) = \frac{\text{Number who died from cancer}}{\text{Total number of individuals}} = \frac{867}{136,961} \approx 0.0063 ]
So, the probability is approximately 0.0063, or 0.63%.
(b) Probability of Being a Current Cigar Smoker:
Similarly, the probability that a randomly selected individual is a current cigar smoker is:
[ P(\text{Current cigar smoker}) = \frac{\text{Number of current cigar smokers}}{\text{Total number of individuals}} = \frac{6,032}{136,961} \approx 0.044 ]
Thus, the probability is approximately 0.044, or 4.4%.
(c) Probability of Dying from Cancer and Being a Current Cigar Smoker:
The probability of both events occurring simultaneously is:
[ P(\text{Died from cancer and current cigar smoker}) = \frac{\text{Number who died from cancer and are current cigar smokers}}{\text{Total number of individuals}} = \frac{193}{136,961} \approx 0.0014 ]
Therefore, the probability is approximately 0.0014, or 0.14%.
(d) Probability of Dying from Cancer or Being a Current Cigar Smoker:
To find the probability of either event occurring, we use the formula:
[ P(\text{Died from cancer or current cigar smoker}) = P(\text{Died from cancer}) + P(\text{Current cigar smoker}) – P(\text{Died from cancer and current cigar smoker}) ]
Plugging in the values:
[ P = 0.0063 + 0.044 – 0.0014 = 0.0499 ]
Thus, the probability is approximately 0.0499, or 4.99%.
Explanation:
- (a) The probability of 0.0063 indicates that, in this study, about 0.63% of individuals died from cancer.
- (b) A probability of 0.044 means that approximately 4.4% of individuals were current cigar smokers.
- (c) The 0.0014 probability suggests that about 0.14% of individuals both died from cancer and were current cigar smokers.
- (d) The combined probability of 0.0499 implies that roughly 4.99% of individuals either died from cancer or were current cigar smokers, or both.
Health Implications:
Cigar smoking is known to pose significant health risks, including an increased likelihood of developing various cancers. Even among those who do not inhale deeply, the risk for cancers of the oral cavity, esophagus, and larynx is elevated. The data suggests a notable association between current cigar smoking and cancer-related mortality.
Conclusion:
The analysis underscores the importance of understanding the health risks associated with cigar smoking. Even minimal exposure can have significant health consequences. It’s crucial to interpret these probabilities within the context of the study’s design and population. For personalized health advice, consulting healthcare professionals is recommended.