COMP 2711 Ex 10

1

a

$$\begin{array}{rl}P(A_1)& =\frac{1}{2}\times \frac{1}{2}+\frac{1}{2}\times 1\\ & =\frac{3}{4}\\ & \\ P(A_2)& =\frac{1}{2}\times \frac{1}{2}+\frac{1}{2}\times 1\\ & =\frac{3}{4}\end{array}$$

b

$$\begin{array}{rl}P(A_2\mid A_1)& =\frac{P(A_2\cap A_1)}{P(A_1)}\\ & =\frac{\frac{3}{4}\times \frac{3}{4}}{\frac{3}{4}}\\ & =\frac{3}{4}\end{array}$$

c

$$\begin{array}{rl}P(A_2\cup A_1)& =P(A_2)+P(A_1)-P(A_2\cap A_1)\\ & =\frac{3}{4}+\frac{3}{4}-\frac{3}{4}\times \frac{3}{4}\\ & =\frac{15}{16}\end{array}$$

d

$$\begin{array}{rl}& P(A_1)P(A_2)=\frac{3}{4}\times \frac{3}{4}=\frac{9}{16}\\ & P(A_1\cap A_2)=\frac{3}{4}\times \frac{3}{4}=\frac{9}{16}\end{array}$$

Since $P(A_1)P(A_2)=P(A_1\cap A_2)$, $A_1$ and $A_2$ are independent