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In probability theory, according to Boole's inequality, also denoted as the union bound, for any finite or countable set of events, the probability that at least one of the events happens is no higher than the sum of the probabilities of the individual events.

In mathematics, the probability theory is denoted as an important branch that studies about the probabilities of the random event. The probability is denoted as the measurement of chances of happening an event which is an outcome of an experiment.

**For Example** − tossing a coin is denoted as an experiment and getting head or tail is denoted as an event. Ideally, there are50%-50% chances, that is 1/2-1/2 probability of obtaining either a head or a tail.

There are so many important concepts in probability theory.

Boole's inequality is one of them.

The union bound or Boole's inequality is applicable when we need to show that the probability of the union of some events is smaller than some value.

Remember that for any two events C and D we have

P(C ∪ D) = P(C) + P(D) − P(C ∩ D) ≤ P(C) + P(D).

Similarly, for three events C, D, and E, we can write

P(C ∪ D ∪ E) = P((C ∪ D) ∪ E) ≤ P(C ∪ D) + P(E) ≤ P(C) + P(D) + P(E).

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