Question

Events A and B are independent. Suppose event A occurs with probability 0.67 and event B occurs with probability 0.70 .

a. If event A or event B occurs, what is the probability that both A and B occur?

b. If B does not occur, what is the probability that A occurs?

Answer 1

Solution:

P(A) = 0.67

P(B) = 0.70

P(A $\cap$ B) = P(A) *P(B) = 0.67 * 0.70 = 0.469

P(A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B) = 0.67 + 0.70 - 0.469 = 0.901

a)

P[(A $\cap$ B) | P(A $\cup$ B)]

= P[(A $\cap$ B) $\cap$ P(A $\cup$ B)] / P(A $\cup$ B)

= P[(A $\cap$ B)] / P(A $\cup$ B)

= 0.469/0.901

= 0.52

a. 0.52

b)

P[A | B^c]

= P[A $\cap$ B^c ] / P[B^c]

= { P(A) - P(A $\cap$ B) } / {1 - P(B) }

= {0.67 - 0.469}/{1 - 0.70}

= 0.67

b. 0.67

Answers:

a. 0.52

b. 0.67