What is the Probability of Road Flooding Given Rain and Sewer Overflow?

[Dustinsfl]

It is known that if it rains, there is a \(50\%\) chance that a sewer will overflow. Also, if the sewer overflows, then there is a \(30\%\) chance that the road will flood. If there is a \(20\%\) chance that it will rain, what is the probability that the road will flood?

Let A be the probability that it will rain, B the probability that the road will flood, and C the probability that the sewer will flood.
What have is then
\[
P[B|A] = \frac{P[A|B]P}{P[A|B]P + P[A|C]P[C]}
\]
However, this is incorrect. The book says the answer is \(0.03\), and I get \(0.375\).
How should the conditional probability be broken up?

 

[Jameson]

It is known that if it rains, there is a \(50\%\) chance that a sewer will overflow. Also, if the sewer overflows, then there is a \(30\%\) chance that the road will flood. If there is a \(20\%\) chance that it will rain, what is the probability that the road will flood?

Let A be the probability that it will rain, B the probability that the road will flood, and C the probability that the sewer will flood.
What have is then
\[
P[B|A] = \frac{P[A|B]P}{P[A|B]P + P[A|C]P[C]}
\]
However, this is incorrect. The book says the answer is \(0.03\), and I get \(0.375\).
How should the conditional probability be broken up?

We are not asked to find $P[B|A]$ because it isn't given that $A$ occurs. We also don't have a way to find $P[A|B]$ and $P[A|C]$ from this info. Using the events you've listed, we are asked to find $P$.

Think of it this way. In order to flood, it must rain, the sewer must overflow and the road must flood (since this doesn't always happen). How can we account for all of these events at once?

 

[Jameson]

Just to wrap up the thread for future reference, the probability of the road flooding can be found by multiplying the three events I mentioned in the post above.

Let $A$ be the probability that it will rain, $B$ be the probability that the sewer will overflow and $C$ be the probability that the road will flood.

$P[ \text{flooding}]=P[A] \cdot P[B|A] \cdot P[C|A,B]=(.2)(.5)(.3) = .03$