Question

A Ph.D. graduate has applied for a job with two universities: A and B. The graduate
feels that she has a 60% chance of receiving an offer from university A and a 50%
chance of receiving an offer from university B. If she receives an offer from
university B, she believes that she has an 80% chance of receiving an offer from
university A.
a. What is the probability that both universities will make her an offer?
b. What is the probability that at least one university will make her an offer?
c. If she receives an offer from university B, what is the probability that she will
not receive an offer from university A?

Answer 1

P(receiving offer from university A) = 0.6

P(receiving offer from university B) = 0.5

P(receiving offer from university A | receives offer from university B) = 0.8

a)  P(receiving offer from university A | receives offer from university B) =$\frac{P(receiving \; offer \; from \; university \; A \; and \; receiving \; offer \; from \; university \; B)}{P(receiving \; offer \; from \; university \; B)}$

$0.8 = \frac{P(receiving \; offer \; from \; university \; A \; and \; receiving \; offer \; from \; university \; B)}{0.5}$

P(receiving offer from university A and receiving offer from university B) = 0.4

b) P(offer from at least one) = P(receiving offer from university A) + P(receiving offer from university B) - P(receiving offer from university A and receiving offer from university B)

P(offer from at least one) = 0.6 + 0.5 - 0.4

P(offer from at least one) = 0.7

c) P(receiving offer from university B and not receiving offer from university A) = P(receiving offer from university B) - P(receiving offer from university A and receiving offer from university B)

P(receiving offer from university B and not receiving offer from university A) = 0.5 - 0.4

P(receiving offer from university B and not receiving offer from university A) = 0.1

P(receiving offer from university B | not receiving offer from university A) =$\frac{P(receiving \; offer \; from \; university \; B \; and \; not \; receiving \; offer \; from \; university \; A)}{P(not \; receiving \; offer \; from \; university \; A)}$

P(receiving offer from university B | not receiving offer from university A) $= \frac{0.1}{1 - 0.6}$

P(receiving offer from university B | not receiving offer from university A) = 0.25