Question

If \(A\) and \(B\) are independent events with \(P(A)=0.6\) and \(P(B)=0.3\), find the following:

(a) \(P(A \cup B)\)
(b) \(P\left(A^{\prime} \cap B\right)\)
(c) \(P\left(A^{\prime} \cup B^{\prime}\right)\)
(d) \(P(A \mid B)\)
(e) \(P\left(B^{\prime} \mid A^{\prime}\right)\)

Answer 1

(a) \(P(A \cup B)=P(A)+P(B)-P(A \cap B)=.6+.3-(.6)(.3)=.72\)

(b) \(P\left(A^{\prime} \cap B\right)=P\left(A^{\prime}\right) P(B)=(1-P(A)) P(B)=(.4)(.3)=.12\)

(c) \(P\left(A^{\prime} \cup B^{\prime}\right)=P\left(A^{\prime}\right)+P\left(B^{\prime}\right)-P\left(A^{\prime} \cap B^{\prime}\right)=.4+.7-(.4)(.7)=.82\)

or \(P\left(A^{\prime} \cup B^{\prime}\right)=P\left((A \cap B)^{\prime}\right)=1-P(A \cap B)=1-.18=.82\).

(d) \(P(A \mid B)=P(A)=.6\)

(e) \(P\left(B^{\prime} \mid A^{\prime}\right)=P\left(B^{\prime}\right)=.7\)