Probability proof with independent events

Question

If events $A_1,...A_n$ are independent (in total), show that $$P\left(\bigcup\limits_{i=1}^nA_i\right)=1-\prod\limits_{i=1}^{n}\left(1-P(A_i)\right).$$

Could someone show how to prove this statement?

Answer 1

Here a verbal proof: the left-handed side represents the probability that at least one of the events $A_1,\dots,A_n$ occurs. The quantity $1-P(A_i)$ represents the probability that $A_i$ does not occur, and by independence the product represents the probability that none of the events occurs. Thus, both sides represent the same probability.