Question

If two events, A and B, are such that P(A) = .5, P(B) = .3, and P(AnB) = 1, find

the following

a. P(A|B)

b. P(B|A)

c. P(A|A UB)

d. P(A|ANB)

e. P(ANB|AUB)

Answer 1

a. P(A|B)

$\frac{P(AnB)}{P(B)}=\frac{0.2}{0.5}=\frac{2}{5}$

b. P(B|A)

$\frac{P(BnA)}{P(A)}=\frac{0.2}{0.3}=\frac{2}{3}$

c. P(A|A UB)

$\frac{P(An(AnB))}{P(AuB)}=\frac{P(A)}{P(AuB)}=\frac{0.5}{0.5+0.3-0.1}=\frac{0.5}{0.7}=\frac{5}{7}$

d. P(A|ANB)

$\frac{P((A)n(P(AnB))}{P(AuB)}=\frac{P(A)}{1-P(AnB)}=\frac{0.5}{1-0.1}=\frac{0.5}{0.9}=\frac{5}{9}$

e. P(AnB|AUB)

$\frac{P((AnB)(AuB))}{P(AuB)}=\frac{P(AnB)}{P(AuB)}=\frac{0.2}{0.5+0.3-0.1}=\frac{0.2}{0.7}=\frac{2}{7}$