Probability Calculation for Men's Health Data
This problem involves calculating the joint probability of two events (being a smoker and having cancer) using the information provided about a population of men.
Given Probabilities
- Let C be the event that a man suffers from cancer. Given: $P(C) = 3\% = 0.03$
- Let S be the event that a man is a smoker. Given: $P(S) = 15\% = 0.15$
- Let S U C be the event that a man is either a smoker or has cancer (or both). Given: $P(S \cup C) = 10\% = 0.10$
Finding Joint Probability (Smoker AND Cancer)
We need to find the probability that a randomly selected man is both a smoker and suffers from cancer, denoted as $P(S \cap C)$.
We use the addition rule for probabilities:
$P(S \cup C) = P(S) + P(C) - P(S \cap C)$
To find $P(S \cap C)$, we rearrange the formula:
$P(S \cap C) = P(S) + P(C) - P(S \cup C)$
Calculation Steps
- Substitute the given values into the rearranged formula:
$P(S \cap C) = 0.15 + 0.03 - 0.10$ - Perform the addition:
$P(S \cap C) = 0.18 - 0.10$ - Perform the subtraction to find the final probability:
$P(S \cap C) = 0.08$
Therefore, the probability that a randomly selected man is both a smoker and suffering from cancer is 0.08.
Important Questions from Probability
Three dice are thrown. What is the probability of getting a sum which is a perfect square?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that all three boys sit together?
