Chapter 1.1, Problem 9E

Roll a fair six-sided die three times. Let $A_1=\left(\text{1 or 2 on the first roll}\right),$ $A_2=\left(\text{3 or 4 on the second roll}\right),$ and $A_3=\left(\text{5 or 6 on the third roll}\right)$.

It is given that $P\left(A_i\right)=\frac{1}{3},i=1,2,3;P\left(A_i\cap A_j\right)={\left(\frac{1}{3}\right)}^2,i\ne j;$ and $P\left(A_1\cap A_2\cap A_3\right)={\left(\frac{1}{3}\right)}^3$.

(a) Use Theorem 1.1-6 to find $P\left(A_1\cup A_2\cup A_3\right)$.

(b) Show that $P\left(A_1\cup A_2\cup A_3\right)=1-{\left(1-\frac{1}{3}\right)}^3$.