The Law of Total Probability

Sometimes the probability of an event $A$ is hard to compute directly but easy to compute separately inside each of a few non-overlapping scenarios. The law of total probability combines these pieces into the overall probability.

If $B$ is an event with $0 < P(B) < 1$, then $B$ and its complement $B^c$ split (partition) the sample space. The law of total probability states

$$P(A) = P(A \mid B)\,P(B) + P(A \mid B^c)\,P(B^c).$$

For example, suppose tomorrow is rainy with probability $0.4$. If it rains, you are late with probability $0.6$; if it does not rain, you are late with probability $0.1$. Let $A = \{\text{late}\}$ and $B = \{\text{rainy}\}$. Then

$$\begin{align*} P(A) &= P(A \mid B)\,P(B) + P(A \mid B^c)\,P(B^c) \\ &= 0.6 \cdot 0.4 + 0.1 \cdot 0.6 \\ &= 0.24 + 0.06 \\ &= 0.30. \end{align*}$$

Think of it as a weighted average of the conditional probabilities, where the weights are the probabilities of the two scenarios.