Answer:
a) 0.685
b) 0.8175
c) 0.254
Step-by-step 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!!!
