Conditional probability question with three independent events

Question

I have the following probabilities:

$$P(A_1)= P(A_2) = P(A_3)= .2,$$ where $A_1, A_2,$ and $A_3$ are all independent.

I wish to find $$P(A_1\mid A_1 \cup A_2 \cup A_3).$$

I know that $$P(A_1 \cup A_2 \cup A_3) = 3(.2) - 3(.2)(.2) + (.2)(.2)(.2) = .488.$$

I do not know how to proceed after this, however. I do not understand how to incorporate the conditional part of the problem into my answer.

Any guidance is much appreciated. Thanks!

Answer 1

Just use the definition of Conditional Probability, and the fact that $A_1\subseteq A_1\cup A_2\cup A_3$.

$$\mathsf P(A_1\mid A_1\cup A_2\cup A_3)=\dfrac{\mathsf P(A_1)}{\mathsf P(A_1\cup A_2\cup A_3)}=\dfrac{0.200}{0.488}$$