The probability of rain falling on the roof of Adam’s dorm on any given day is 0.15. The probability of rain falling on the lecture hall is 0.20. On days when rain strikes the dorm roof, it has a 0.90 probability of also striking the lecture hall roof. On a day when rain does not strike the dorm roof, what is the probability of rain striking the lecture hall roof?
To find the probability of rain striking the lecture hall roof on a day when rain does not strike the dorm roof, we can use the concept of conditional probability.
Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, the event that we are interested in is rain striking the lecture hall roof, and the condition is rain not striking the dorm roof.
Let's denote the events:
A = Rain strikes the dorm roof
B = Rain strikes the lecture hall roof
We are given the following information:
P(A) = 0.15 (Probability of rain striking the dorm roof)
P(B) = 0.20 (Probability of rain striking the lecture hall roof)
P(B|A) = 0.90 (Probability of rain striking the lecture hall roof given that rain strikes the dorm roof)
We want to find P(B|A') (Probability of rain striking the lecture hall roof given that rain does not strike the dorm roof).
Using the definition of conditional probability, we can write:
P(B|A') = P(B and A') / P(A')
To find P(B and A'), we can use the fact that the events A and B are not independent. Therefore, we need to use the joint probability formula:
P(B and A') = P(A') * P(B|A')
Since A' is the complement of A (i.e., rain does not strike the dorm roof), P(A') = 1 - P(A) = 1 - 0.15 = 0.85.
Substituting the values, we have:
P(B and A') = 0.85 * 0.90 = 0.765
Now, we need to find P(A'):
P(A') = 1 - P(A) = 1 - 0.15 = 0.85
Finally, substituting the values into the conditional probability formula:
P(B|A') = P(B and A') / P(A') = 0.765 / 0.85 ≈ 0.9
Therefore, the probability of rain striking the lecture hall roof on a day when rain does not strike the dorm roof is approximately 0.9 or 90%.