addition rule for triple

The addition rule for a triple union probability is similar to the foregoing rule.

For any three events $A,B$ , and $C$ ,

$$\begin{array}{rl}P\left(A\cup B\cup C\right)=& P\left(A\right)+P\left(B\right)+P\left(C\right)\\ & -P\left(A\cap B\right)-P\left(A\cap C\right)-P\left(B\cap C\right)\\ & +P\left(A\cap B\cap C\right)\end{array}$$

---

This can be verified by examining the Venn diagram of $A\cup B\cup C$ , shown in Figure 2.6.

Figure 2.6 $A\cup B\cup C$

When $P\left(A\right),P\left(B\right)$ , and $P\left(C\right)$ are added, the intersection probabilities $P\left(A\cap B\right)$ , $P\left(A\cap C\right)$ , and $P\left(B\cap C\right)$ are all counted twice. Each one must therefore be subtracted. But then $P\left(A\cap B\cap C\right)$ has been added in three times and subtracted out three times, so it must be added back. In general, the probability of a union of $k$ events is obtained by summing individual event probabilities, subtracting double intersection probabilities, adding triple intersection probabilities, subtracting quadruple intersection probabilities, and so on.

---