Question

A and B are events with Pr[AB] = 0.5, PrIAB] = 0.3, PrIABI = 0.1, and Pr[ABl = 0.1. a. What are Pr[Al and Pr[B]? b. What are PrA B PrlAB], Pr[A|B], and PrAB? c. Calculate Pr[B|A] from Pr[A B using Bayes theorem. 2.11

Answer 1

a)

$P(A)=P(A\overline{B})+P(AB)=0.3+0.5=0.8$

and, $P(B)=P(\overline{A}B)+P(AB)=0.1+0.5=0.6$

b)

$P(A|B)=\frac{P(AB)}{P(B)}=\frac{0.5}{0.6}=0.8333$

$P(A|\overline{B})=\frac{P(A\overline{B})}{P(\overline{B})}=\frac{0.3}{1-0.6}=0.75$

$P(\overline{A}|B)=\frac{P(\overline{A}B)}{P(B)}=\frac{0.1}{0.6}=0.1667$

$P(\overline{A}|\overline{B})=\frac{P(\overline{A}\overline{B})}{P(\overline{B})}=\frac{0.1}{1-0.6}=0.25$

c)

$P(B|A)=\frac{P(B).P(A|B)}{P(B).P(A|B)+P(\overline{B}).P(A|\overline{B})}=\frac{0.6(0.8333)}{0.6(0.8333)+(1-0.6)0.75}=0.625$

$P(A|B)=\frac{P(A).P(B|A)}{P(A).P(B|A)+P(\overline{A}).P(B|\overline{A})}=\frac{0.8(0.625)}{0.8(0.625)+0.1}=0.8333$