Question

You want to go to graduate school, so you ask your math professor, Dr. Emmy Noether, for a letter of recommendation. You estimate that there is a 80% chance that you will get into a graduate program if you receive a strong recommendation, a 60% chance that you will get into a graduate program if you receive a moderately good recommendation, and 5% chance that you will get into a graduate program if you receive a weak recommendation. Furthermore, you estimate that the probabilities that a recommendation will be strong, moderately good, and weak are 0.7, 0.2, and 0.1, respectively.(a) Based on these estimates, what is the probability that you will get into a graduate pro-gram? (b) Given that you did receive an offer to attend a graduate program, what is the probability that you received a strong recommendation? (c) Suppose you didn't receive an offer to attend a graduate program. Given that, what is the probability that you received a moderately good recommendation?

Answer 1

a) 0.685

b) 0.8175

c) 0.254

Explanation:

Let the event that one gets into graduate school be G

The event that one does not get into graduate school is G'

Let the event that one gets a strong recommendation be S

Let the event that one gets a moderately good recommendation be M

Let the event that one gets a weak recommendation be W

P(G|S) = 80% = 0.80

P(G|M) = 60% = 0.60

P(G|W) = 5% = 0.05

P(S) = 0.7

P(M) = 0.2

P(W) = 0.1

These 3 probabilities add up to a 1.0, so, it means these are all the possible outcomes for seeking a recommendation.

a) Probability that you will get into a graduate program = P(G)

P(G) = P(G n S) + P(G n M) + P(G n W)

But the conditional probability P(A|B) is given as

P(A|B) = P(A n B) ÷ P(B)

P(A n B) = P(A|B) × P(B)

Hence,

P(G n S) = P(G|S) × P(S) = 0.80 × 0.70 = 0.56

P(G n M) = P(G|M) × P(M) = 0.6 × 0.2 = 0.12

P(G n W) = P(G|W) × P(W) = 0.1 × 0.05 = 0.005

P(G) = P(G n S) + P(G n M) + P(G n W)

P(G) = 0.56 + 0.12 + 0.005 = 0.685

b) Given that one does receive an offer, probability that you received a strong recommendation?

This probability = P(S|G)

P(S|G) = P(G n S) ÷ P(G) = 0.56 ÷ 0.685 = 0.8175

c) Suppose you didn't receive an offer to attend a graduate program. Given that, what is the probability that you received a moderately good recommendation?

Probability that one doesn't get the offer, given one got a strong recommendation = P(G'|S)

P(G'|S) = 1 - P(G|S) = 1 - 0.80 = 0.20

Probability that one doesn't get job offer, given one got a moderate recommendation = P(G'|M)

P(G'|M) = 1 - P(G|M) = 1 - 0.60 = 0.40

Probability that one doesn't get job offer, given one got a weak recommendation = P(G'|S)

P(G'|W) = 1 - P(G|W) = 1 - 0.05 = 0.95

Total probability that one doesn't get the offer

P(G') = P(G' n S) + P(G' n M) + P(G' n W)

P(G' n S) = P(G'|S) × P(S) = 0.20 × 0.70 = 0.14

P(G' n M) = P(G'|M) × P(M) = 0.40 × 0.20 = 0.08

P(G' n W) = P(G'|W) × P(W) = 0.95 × 0.10 = 0.095

Total probability that one doesn't get the offer

P(G') = P(G' n S) + P(G' n M) + P(G' n W)

= 0.14 + 0.08 + 0.095 = 0.315

Given that one does not receive the job offer, the probability that you received a moderately good recommendation

This probability = P(M|G')

P(M|G') = P(G' n M) ÷ P(G') = 0.08 ÷ 0.315 = 0.254

Hope this Helps!!!