Question

Q2

if two events, A and B, are such that P(A) = .5, P(B) = .3, and P(A intersection B) = .1, findthe following:

a).P(A|B)

b). P(B|A)

c).P(A|AUB)

d).P(A|A intersection B)

e). P(A intersection B|A U B)

Answer 1

a).

$P(A|B)=\frac{P(A\bigcap B)}{P(B)}\\ P(A\bigcap B)=0.1\\ P(B)=0.3\\ P(A|B)=\frac{0.1}{0.3}=\frac{1}{3}\\$

b).

$P(B|A)=\frac{P(B\bigcap A)}{P(A)}\\ P(B\bigcap A)=P(A\bigcap B)=0.1\\ P(A)=0.5\\ P(B|A)=\frac{0.1}{0.5}=\frac{1}{5}\\$

c).

$P(A|A\bigcup B)=\frac{P(A\bigcap (A\bigcup B))}{P(A\bigcup B)}\\ P(A|A\bigcup B)=\frac{P(A)}{P(A\bigcup B)}\\ P(A)=0.5\\ P(A\bigcup B)=P(A)+P(B)-P(A\bigcap B)\\ P(A\bigcup B)=0.5+0.3-0.1=0.7\\ P(A|A\bigcup B)=\frac{0.5}{0.7}=\frac{5}{7}\\$

d).

$P(A|A\bigcap B)=\frac{P(A\bigcap (A\bigcap B))}{P(A\bigcap B)}\\ P(A|A\bigcap B)=\frac{P(A\bigcap B)}{P(A\bigcap B)}=1\\$

e).

$P(A\bigcap B|A \bigcup B)=\frac{P(A\bigcap B\bigcap A \bigcup B)}{P(A \bigcup B)}\\ P(A\bigcap B|A \bigcup B)=\frac{P(A\bigcap B)}{P(A \bigcup B)}\\ P(A\bigcap B|A \bigcup B)=\frac{0.1}{0.7}=\frac{1}{7}\\$