Statistics for Business: Decision Making and Analysis, 3rd Edition

... nontechnical language. This calculation relies on past performance of the assembly line; if things have changed, the probability might be too high or too low. The calculation also relies on independence from day to day.

Boole’s Inequality

Probabilities of unions are hard to find unless the sets are disjoint. One needs to adjust for double counting, requiring the probability of intersections. These difficulties compound for unions of many sets, as in the prior example of breakdowns in five days. If the probabilities are small, however, a simple upper bound for the probability of the union is often very accurate.

Boole’s Inequality

$$P\left({\text{A}}_{1}or{\text{A}}_{2}or\dots or{\text{A}}_k\right)\le p_1+p_2+\dots +p_k$$

7.3-9 Full Alternative Text

Boole’s inequality (sometimes called Bonferroni’s inequality) ...