The union bound states that P(⋃ni=1Ai)≤∑ni=1P(Ai) for any events A1,…,An. In practice, this is an enormously useful property. I think one reason it’s so useful is that for rare events, it gives you essentially the same answer you’d get if the events were independent, but without proving anything about their dependence structure. To be precise, suppose A1,…,An are independent and all have probability p. Then, P(⋃ni=1Ai)=1−(1−p)n and ∑ni=1P(Ai)=np. Now note that 1−(1−p)n∈Θ(np) when np is bounded, so the union bound gives you asymptotically the same bound as assuming the events are independent.
The union bound states that P(⋃ni=1Ai)≤∑ni=1P(Ai) for any events A1,…,An. In practice, this is an enormously useful property. I think one reason it’s so useful is that for rare events, it gives you essentially the same answer you’d get if the events were independent, but without proving anything about their dependence structure. To be precise, suppose A1,…,An are independent and all have probability p. Then, P(⋃ni=1Ai)=1−(1−p)n and ∑ni=1P(Ai)=np. Now note that 1−(1−p)n∈Θ(np) when np is bounded, so the union bound gives you asymptotically the same bound as assuming the events are independent.