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

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The probability of road flooding is calculated by considering the chances of rain, sewer overflow, and road flooding. Given a 20% chance of rain, a 50% chance of sewer overflow if it rains, and a 30% chance of road flooding if the sewer overflows, the overall probability of flooding is determined by multiplying these probabilities together. The correct calculation yields a probability of 0.03 for road flooding. This approach clarifies that the events must occur sequentially for flooding to happen. Understanding the conditional probabilities is crucial for accurate calculations in such scenarios.
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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?
 
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dwsmith said:
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?
 
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.

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