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Summary

The law of total probability states that the total probability of an outcome which can be realized via several distinct events can be expressed by the sum of the probabilities of each event. This is demonstrated by the formula $P(A) = \sum_{i=1}^n P(A \cap B_i)$.
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This idea is also expressed by looking at a partition of $S$, and adding the amount of probability of $A$ that falls in each.
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According to

Summary
In probability theory , the law (or formula ) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities . It expresses the total probability of an outcome which can be realized via several distinct events , hence the name.

Law of total probability - Wikipedia
wikipedia.org

Summary
This is the idea behind the law of total probability, in which the area of forest is replaced by probability of an event $A$ . In particular, if you want to find $P(A)$, you can look at a partition of $S$, and add the amount of probability of $A$ that falls in each

Law of Total Probability | Partitions | Formulas
probabilitycourse.com

to be particularly useful when it was combined with the chain rule and gave rise to a tool so useful, it was given the big name, law of total probability.

Law of Total Probability
chrispiech.github.io