Question

16. Given: ?(?) = 0.45, ?(?) = 0.65, and ?(? ??? ?) = 0.35. Find

(a) Construct the Venn Diagram.

(b) ?(? ?? B)

(c) ?(?̅̅̅??̅̅̅̅̅?̅)

(d) ?(?|?)

(e) ?(?|?)

Answer 1

Given: P(A) = 0.45, P(B) = 0.65, P(A and B) =0.35

#a) Venn diagram

U =1 , we have total P( A or B) = 0.75 , so outside of A and B. We mention 0.25.

#b) P(A or B) = P(A) +P(B) – P(A and B)

= 0.45 +0.65 -0.35

P(A or B ) = 0.75

#c) P(A’ or B’)      ( please read A’ as A compliment )

Using Demorgan’s law : P(A’ or B’) = (A and B )’   ..............(1)

We know P(A and B) = 0.35

P(A and B) ‘ =1 – P(A and B)

P(A and B) ‘ =1 – 0.35

P(A and B) ‘ =0.65

P(A’ or B’) = (A and B )’      ...      by (1)

P(A’ or B’) = 0.65

#d)P(A/B)

First we will check A and B are independent or dependent...

If P(A) *P(B) = P( A and B ) ,then, we say A and B are independent.

P(A) *P(B) = 0.45 *0.65 = 0.292

P( A and B) = 0.35

As , P(A) *P(B) ≠ P( A and B ) , A and B are dependent.

P(A / B) = P( A and B) / P (B)

=0.35/0.65

P(A / B ) =0.538

#e) P(B/ A) = P( A and B) / P (A)

=0.35/0.45

P(B / A ) =0.777