What is the probability that it is raining?

Question

A bus comes around your house every day. When it rains, there is a 20% probability that it is going to be late. When it doesn't rain, there is a 10% probability that it's going to be late. The probability that it will rain is 40%. Today, the bus was late. What is the probability that it was raining? (By the way, the correct answer is 0.571, but I couldn't figure out how to get there.)

Answer 1

According to Bayes' Rule,

$$ P[A|B] = \frac{P[B|A]\cdot P[A]}{P[B]} $$

This gives us

$$ P[\text{Rain}|\text{Late}] = \frac{P[\text{Late}|\text{Rain}]\cdot P[\text{Rain}]}{P[\text{Late}]} $$

Of these, we know

$$ P[\text{Late}|\text{Rain}] = 0.2 , $$ $$ P[\text{Rain}] = 0.4 . $$

So, we still need $P[\text{Late}]$.

For this, we can use the fact that

$$ P[A] = P[A \cap B]+P[A \cap B^C] . $$

This gives us

$$ P[\text{Late}]=P[\text{Late} \cap \text{Rain}]+P[\text{Late} \cap \text{No Rain}] $$

$$ =P[\text{Late} | \text{Rain}]\cdot P[\text{Rain}]+P[\text{Late} | \text{No Rain}]\cdot P[\text{No Rain}] $$

$$ = 0.2 \cdot 0.4 + 0.1 \cdot (1-0.4) $$ $$ = 0.14 $$

Substituting these values all into the expression we found above, we have:

$$ P[\text{Rain}|\text{Late}] = \frac{P[\text{Late}|\text{Rain}]\cdot P[\text{Rain}]}{P[\text{Late}]} $$

$$ = \frac{0.2 \cdot 0.4}{0.14} $$ $$ \approx 0.571 $$

Answer 2

We use the rules of conditional probability. Generally speaking: $$\mathbb{P}(\text{A}|\text{B}) = \frac{\mathbb{P}(\text{A} \text{ and } \text{B})}{\mathbb{P}(\text{B})}$$ Where $\mathbb{P}(\text{A}|\text{B})$ means "probability of A given B".

So here we have the probability that it is raining is $$\frac{\text{probability that it is raining and the bus was late}}{\text{probability that the bus was late}}$$ The probability that the bus was late is $0.4 \times 0.2 + 0.6 \times 0.1$. (The sum of the case the bus is late and it rains and the bus is late and it does not rain.

The probability that the bus was late and it is raining is $0.4 \times 0.2$

So the answer is $0.4\times0.2/(0.4\times0.2+0.6\times0.1) = 0.571$