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 questions based on the provided data:
Data Summary:
| Smoking Status | Died from Cancer | Did Not Die from Cancer | Total |
|---|---|---|---|
| Never smoked cigars | 620 | 123,094 | 123,714 |
| Former cigar smoker | 54 | 7,161 | 7,215 |
| Current cigar smoker | 193 | 5,839 | 6,032 |
| Total | 867 | 136,094 | 136,961 |
(a) Probability that an individual died from cancer:
The probability is calculated by dividing the total number of individuals who died from cancer by the grand total of individuals:
[ P(\text{Died from Cancer}) = \frac{867}{136,961} \approx 0.0063 ]
This means there’s approximately a 0.63% chance that a randomly selected individual from this study died from cancer.
(b) Probability that an individual was a current cigar smoker:
Similarly, the probability is:
[ P(\text{Current Cigar Smoker}) = \frac{6,032}{136,961} \approx 0.044 ]
So, there’s about a 4.4% chance that a randomly selected individual was a current cigar smoker.
(c) Probability that an individual died from cancer and was a current cigar smoker:
This probability is found by dividing the number of individuals who both died from cancer and were current cigar smokers by the grand total:
[ P(\text{Died from Cancer and Current Cigar Smoker}) = \frac{193}{136,961} \approx 0.0014 ]
Thus, there’s about a 0.14% chance that a randomly selected individual both died from cancer and was a current cigar smoker.
(d) Probability that an individual died from cancer or was a current cigar smoker:
For this, we use the formula:
[ P(A \cup B) = P(A) + P(B) – P(A \cap B) ]
Applying the values:
[ P(\text{Died from Cancer or Current Cigar Smoker}) = 0.0063 + 0.044 – 0.0014 = 0.0499 ]
Therefore, there’s about a 4.99% chance that a randomly selected individual either died from cancer or was a current cigar smoker.
Explanation:
These calculations are based on the principles of probability, particularly when dealing with contingency tables. A contingency table displays the frequency distribution of variables, allowing for the calculation of joint, marginal, and conditional probabilities. By organizing data in this manner, we can easily determine the likelihood of various events and their interrelations.