[Math] Occurrence of at least one event.

Question

I am new to probability and its concepts trying to answer the following question.

Let $F, G$ and $H$ be Pairwise Independent events such that $P(F) = P(G) = P(H) = 1/3$ and $P(F \& G \& H) = 1/4$. Then, the probability that at least one event among $F, G$ and $H$ occurs.

I understand that $P(\text{Event occurring at least once}) = 1 – P(\text{Event not occurring at all})$ but I am failing to apply this in the question above.

Thank you for your help.

Answer 1

The event that at least one of $F,G,H$ occur is equivalent to $F\cup G\cup H$ (that is, only one of the three possible events occur, or any two of the events occur, or all three occur simultaneously).

By the inclusion-exlusion principle, we have

$$\mathbb P(F\cup G\cup H)=\mathbb P(F)+\mathbb P(G)+\mathbb P(H)-\mathbb P(F\cap G)-\mathbb P(G\cap H)-\mathbb P(F\cap H)+\mathbb P(F\cap G\cap H)$$

Given that $F,G,H$ are pairwise independent, we know that

$$\mathbb P(F\cap G)=\mathbb P(F)\mathbb P(G)$$ $$\mathbb P(F\cap H)=\mathbb P(F)\mathbb P(H)$$ $$\mathbb P(G\cap H)=\mathbb P(G)\mathbb P(H)$$

and we know $\mathbb P(F\cap G\cap H)$, so we have everything we need to show that

$$\mathbb P(F\cup G\cup H)=\frac{11}{12}$$