MATH 2411 Homework 1

Q1

$$\begin{array}{rl}P(A\cap C\mid B)& =\frac{P(A\cap C\cap B)}{P(B)}\\ & =\frac{P(A)P(B\mid A)P(C\mid B)}{P(B)}\\ & =\frac{P(B\mid A)P(A)}{P(B)}P(C\mid B)\\ & =P(A\mid B)P(C\mid B)\end{array}$$

Q2

Please find attached at the end of this document

Q3

a

$$\begin{array}{rl}P(A\cap B\cap C)& =P(C\mid A\cap B)P(A\cap B)\\ & =0.2\times P(B\mid A)P(A)\\ & =0.2\times 0.75\times 0.3\\ & =0.045\end{array}$$

b

$$\begin{array}{rl}P(B^c\cap C)& =P(B^c\cap C\cap A)+P(B^c\cap C\cap A^c)\\ & =P(C\mid A\cap B^c)P(A\cap B^c)+P(C\mid A^c\cap B^c)P(A^c\cap B^c)\\ & =0.8\times P(B^c\mid A)P(A)+0.9\times P(B^c\mid A^c)P(A^c)\\ & =0.8\times (1-0.75)\times 0.3+0.9\times (1-0.2)\times (1-0.3)\\ & =0.564\end{array}$$

c

$$\begin{array}{rl}P(B^c\cap C)& =P(C)-P(B\cap C)\\ P(C)& =P(B^c\cap C)+P(B\cap C)\\ & =0.564+P(B\cap C\cap A)+P(B\cap C\cap A^c)\\ & =0.564+0.045+P(C\mid A^c\cap B)P(A^c\cap B)\\ & =0.564+0.045+0.15\times P(B\mid A^c)P(A^c)\\ & =0.564+0.045+0.15\times 0.2\times (1-0.3)\\ & =0.63\end{array}$$

d

$$\begin{array}{rl}P(A\mid C\cap B^c)& =\frac{P(C\cap B^c\mid A)P(A)}{P(C\cap B^c)}\\ & =\frac{\frac{P(C\cap B^c\cap A)}{P(A)}P(A)}{P(C\cap B^c)}\\ & =\frac{P(C\cap B^c\cap A)}{0.564}\\ & =\frac{P(C\mid A\cap B^c)P(A\cap B^c)}{0.564}\\ & =\frac{0.8\times P(B^c\mid A)P(A)}{0.564}\\ & =\frac{0.8\times (1-0.75)\times 0.3}{0.564}\\ & =0.106382978\end{array}$$