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 break down the information in the given table:
| Status | 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 |
Now, let’s answer each part of the question.
(a) Probability that an individual died from cancer:
First, we need to find the total number of individuals who died from cancer and the total number of individuals in the study.
Total number of individuals who died from cancer:
[
\text{Died from cancer} = 620 + 54 + 193 = 867
]
Total number of individuals in the study:
[
\text{Total number of individuals} = (620 + 123,094) + (54 + 7,161) + (193 + 5,839) = 137,061
]
So, the probability that an individual died from cancer is:
[
P(\text{Died from cancer}) = \frac{867}{137,061} \approx 0.0063
]
(b) Probability that an individual was a current cigar smoker:
To find the probability that an individual was a current cigar smoker, we need the total number of current cigar smokers and the total number of individuals in the study.
Total number of current cigar smokers:
[
\text{Current cigar smoker} = 193 + 5,839 = 6,032
]
The probability that an individual is a current cigar smoker is:
[
P(\text{Current cigar smoker}) = \frac{6,032}{137,061} \approx 0.044
]
(c) Probability that an individual died from cancer and was a current cigar smoker:
This is the probability of the intersection of two events: dying from cancer and being a current cigar smoker. From the table, the number of individuals who died from cancer and were current cigar smokers is 193.
Thus, the probability that an individual died from cancer and was a current cigar smoker is:
[
P(\text{Died from cancer and current cigar smoker}) = \frac{193}{137,061} \approx 0.0014
]
(d) Probability that an individual died from cancer or was a current cigar smoker:
To find the probability of either event happening, we can use the formula for the union of two events:
[
P(A \cup B) = P(A) + P(B) – P(A \cap B)
]
Where:
- ( A ) is the event that the individual died from cancer.
- ( B ) is the event that the individual was a current cigar smoker.
- ( P(A \cap B) ) is the probability that the individual died from cancer and was a current cigar smoker.
We already know:
- ( P(A) = \frac{867}{137,061} )
- ( P(B) = \frac{6,032}{137,061} )
- ( P(A \cap B) = \frac{193}{137,061} )
Thus:
[
P(\text{Died from cancer or current cigar smoker}) = \frac{867}{137,061} + \frac{6,032}{137,061} – \frac{193}{137,061}
]
[
P(\text{Died from cancer or current cigar smoker}) = \frac{6,706}{137,061} \approx 0.049
]
Summary of Answers:
(a) Probability that an individual died from cancer: 0.0063
(b) Probability that an individual was a current cigar smoker: 0.044
(c) Probability that an individual died from cancer and was a current cigar smoker: 0.0014
(d) Probability that an individual died from cancer or was a current cigar smoker: 0.049
Explanation:
The total probability for each event is found by dividing the number of favorable outcomes by the total number of individuals in the study, which gives us the proportion of the population affected by each condition.
- Part (a) looks at the probability of an individual dying from cancer, regardless of smoking status.
- Part (b) asks for the probability of being a current cigar smoker, which is found by focusing on the group of current cigar smokers.
- Part (c) combines the two conditions, so we find the probability that both events occur (dying from cancer and smoking cigars).
- Part (d) applies the principle of inclusion and exclusion to calculate the probability that either event happens (dying from cancer or being a current cigar smoker), accounting for any overlap between the two groups (those who both died from cancer and are current cigar smokers). This method ensures we don’t double-count individuals who meet both criteria.
Each part of the problem uses basic principles of probability, such as calculating individual probabilities and applying the union and intersection formulas.