Question

Question 1.2 [2, 2, 2]

Suppose that we have two events A and B such that P A( ) 0.8 = and P B( ) 0.7 = .

a) Is it possible that PA B ( ) 0.1 ∩ = ? Explain your answer.

b) What is the smallest possible value of PA B ( ) ∩ ?

c) What is the largest possible value of PA B ( ) ∩ ?

Answer 1

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

$$P(A\cap B)=P(A)+P(B)-P(A\cup B)$$

Since $P(A)=0.8, P(B)=0.7,0\le P(A\cup B)\le1,$

$P(A\cap B)\le \min{(P(A), P(B))}$ then

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

$$0\le P(A)+P(B)-P(A\cap B)\le 1$$

$$0\le 0.8+0.7-P(A\cup B)\le 1$$

$$P(A\cap B)\le P(B), P(A\cap B)\ge 08+0.7-1$$

$$0.5\le P(A\cap B)\le 0.7$$

a) It is impossible that $P(A\cap B)=0.1.$

b) The smallest possible value of $P(A\cap B)$ is $0.5.$

c) The largest possible value of $P(A\cap B)$ is $0.7.$