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 answer these questions, we need to work with the given data in the table, which outlines the number of individuals who died from cancer and did not die from cancer, grouped by their smoking status. Let’s first summarize the data provided in a clearer form:
| 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 |
Now, let’s answer each question step-by-step.
(a) Probability that the individual died from cancer:
To find the probability that a randomly selected individual died from cancer, we need to divide 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 that a randomly selected individual died from cancer is approximately 0.0063, or 0.63%.
(b) Probability that the individual was a current cigar smoker:
To find the probability that a randomly selected individual was a current cigar smoker, we divide the number of current cigar smokers by the total number of individuals.
[
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
]
So, the probability that a randomly selected individual is a current cigar smoker is approximately 0.044, or 4.4%.
(c) Probability that the individual died from cancer and was a current cigar smoker:
This probability can be calculated by dividing the number of individuals who both died from cancer and were current cigar smokers by the total number of individuals.
[
P(\text{Died from cancer and was a current cigar smoker}) = \frac{\text{Number who died from cancer and were current cigar smokers}}{\text{Total number of individuals}} = \frac{193}{136,961} \approx 0.0014
]
So, the probability that a randomly selected individual died from cancer and was a current cigar smoker is approximately 0.0014, or 0.14%.
(d) Probability that the individual died from cancer or was a current cigar smoker:
To find the probability that a randomly selected individual either died from cancer or was a current cigar smoker, we use the formula for the union of two events:
[
P(\text{Died from cancer or was a current cigar smoker}) = P(\text{Died from cancer}) + P(\text{Current cigar smoker}) – P(\text{Died from cancer and was a current cigar smoker})
]
We already have the individual probabilities from previous calculations, so:
[
P(\text{Died from cancer or was a current cigar smoker}) = 0.0063 + 0.044 – 0.0014 = 0.0499
]
So, the probability that a randomly selected individual either died from cancer or was a current cigar smoker is approximately 0.0499, or 4.99%.
Explanation:
In this problem, we are calculating probabilities based on the frequency of specific events occurring within a population. By dividing the number of favorable outcomes by the total number of possible outcomes, we can estimate the likelihood of certain events.
- For part (a), we are finding the likelihood of death from cancer, which is just the proportion of people who died from cancer.
- For part (b), we are calculating the probability of being a current cigar smoker, which is the proportion of current cigar smokers.
- In part (c), we are interested in the intersection of two events, so we compute the likelihood of both events happening together (i.e., death from cancer and current cigar smoking).
- Finally, part (d) deals with the union of two events, meaning we are calculating the probability that at least one of the two events occurs: either the individual died from cancer or is a current cigar smoker.
These calculations help us understand the relationships between different conditions and their probabilities within the context of the study.