The probability of event A occurring is 0.3, while the probability that event B occurs is 0.8. The probability that event A and event B occur simultaneously is 0.2. If it is known that event A occurred, what is the probability that event B occurred also?
The correct answer and explanation is :
To solve this problem, we need to calculate the conditional probability that event B occurs given that event A has already occurred. This is represented mathematically as P(B∣A)P(B | A), which is the probability of event B occurring given that event A has occurred.
The formula for conditional probability is: P(B∣A)=P(A∩B)P(A)P(B | A) = \frac{P(A \cap B)}{P(A)}
Where:
- P(A∩B)P(A \cap B) is the probability that both event A and event B occur simultaneously.
- P(A)P(A) is the probability that event A occurs.
From the problem, we are given the following values:
- P(A)=0.3P(A) = 0.3
- P(B)=0.8P(B) = 0.8
- P(A∩B)=0.2P(A \cap B) = 0.2
We are asked to find P(B∣A)P(B | A), the probability of event B occurring given that event A has occurred. Plugging the known values into the formula: P(B∣A)=P(A∩B)P(A)=0.20.3P(B | A) = \frac{P(A \cap B)}{P(A)} = \frac{0.2}{0.3} P(B∣A)=23≈0.6667P(B | A) = \frac{2}{3} \approx 0.6667
Thus, the probability that event B occurs given that event A has occurred is approximately 0.6667 or 66.67%.
Explanation:
Conditional probability describes the likelihood of an event occurring given that another event has already occurred. In this case, we are given the probability of event A occurring, the probability of both events A and B occurring together, and the probability of event B occurring independently. Using these values, we apply the conditional probability formula to find how the occurrence of event A affects the likelihood of event B happening. The result tells us that when we know event A has occurred, there is a 66.67% chance that event B will also occur.