Conditional probability

It may be useful to introduce some symbols. Let $L$ be the event she is late for the bus, and let $R$ be the event it rains. We want the probability that it rains, given that she is late. In symbols, we want $\Pr(R|L)$.

There are some useful formulas that come into play. The most basic one comes from the definition of conditional probability. We have $$\Pr(R|L)\Pr(L)=\Pr(R\cap L).$$ If we can calculate $\Pr(L)$ and $\Pr(R\cap L)$ we will be finished.

We first compute $\Pr(R\cap L)$. The probability it rains is $\frac{1}{4}$. Given that it rains, the probability she is late is $\frac{2}{3}$. So the probability it rains and she is late is $\frac{1}{4}\cdot\frac{2}{3}$.

Next we compute $\Pr(L)$. She can be late in two ways: (i) It rains and she is late or (ii) It doesn't rain and she is late. We already computed the probability of (i). In the same way, we find that the probability it doesn't rain and she is late is $\frac{3}{4}\cdot\frac{1}{5}$. To find $\Pr(L)$, add together the probabilities of (i) and of (ii).

Remark: Here is an imprecise but useful way to view the calculation. Imagine we track what happens over $300$ days. It will rain about $75$ days. On two-thirds of these days she will be late, so she will be late on about $50$ days because of rain. It will not rain about $225$ days, and she will be late on about one-fifth of these days, so $45$ days.

So she is late a total of $50+45$ days. Let's confine attention (cut down the sample space) to days she was late. On $50$ of those days it was raining. So the probability it is raining given that she is late should be about $\frac{50}{95}$.

This is an application of the Bayes' theorem: $$ \mathbb{P}(\text{rain}| \text{on time}) = \frac{\mathbb{P}(\text{rain} \land \text{on time} )}{\mathbb{P}(\text{on time} )} = \frac{\mathbb{P}( \text{on time} | \text{rain} ) \mathbb{P}(\text{rain})}{\mathbb{P}(\text{on time} | \text{rain} ) \mathbb{P}(\text{rain}) + \mathbb{P}(\text{on time} | \text{no rain} ) \mathbb{P}(\text{no rain})} = \frac{\frac{1}{3} \cdot \frac{1}{4} }{ \frac{1}{3} \cdot \frac{1}{4} + \frac{4}{5} \cdot \frac{3}{4} } = \frac{5}{41} $$