A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that there is an 80 percent chance that she will get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation. She further estimates that the probabilities that the recommendation will be strong, moderate, and weak are 0.7, 0.2, and 0.1, respectively.

a. How certain is she that she will receive the new job offer?
b. Given that she does receive the offer, how likely should she feel that she received a strong recommendation?

i. a moderate recommendation?
ii. a weak recommendation?

c. Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation?
i. a moderate recommendation?
ii. a weak recommendation?

a) P(Job offer) = 0.65

b) P(S|J) = 0.862

P(M|J) = 0.143

P(W|J) = 0.0179

c) P(S|J') = 0.400

P(M|J') = 0.343

P(W|J') = 0.257

Step-by-step explanation:

- Let the event that she gets a job be J

- Let the event that she does not get the job be J'

- Let the event that she receives a strong recommendation be S

- Let the event that she receives a moderate recommendation be M

- Let the event that she receives a weak recommendation be W.

Given in the question,

P(J|S) = 80% = 0.8

P(J|M) = 40% = 0.4

P(J|W) = 10% = 0.1

P(S) = 0.70

P(M) = 0.20

P(W) = 0.10

a) How certain is she that she will receive the new job offer?

P(J) = P(J n S) + P(J n M) + P(J n W) (since S, M and W are all of the possible outcomes that lead to a job)

But note that the conditional probability, P(A|B) is given mathematically as,

P(A|B) = P(A n B) ÷ P(B)

P(A n B) is then given as

P(A n B) = P(A|B) × P(B)

So,

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

P(J n M) = P(J|M) × P(M) = 0.40 × 0.20 = 0.08

P(J n W) = P(J|W) × P(W) = 0.10 × 0.10 = 0.01

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

P(J) = 0.56 + 0.08 + 0.01 = 0.65

b) Given that she does receive the offer, how likely should she feel that she received a strong recommendation?

This probability = P(S|J)

P(S|J) = P(J n S) ÷ P(J) = 0.56 ÷ 0.65 = 0.862

i. a moderate recommendation?

P(M|J) = P(J n M) ÷ P(J) = 0.08 ÷ 0.65 = 0.143

ii. a weak recommendation?

P(W|J) = P(J n W) ÷ P(J) = 0.01 ÷ 0.65 = 0.0179

c) Probability that she doesn't get job offer, given she got a strong recommendation = P(J'|S)

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

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

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

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

P(J'|W) = 1 - P(J|W) = 1 - 0.10 = 0.90

Total probability that she doesn't get job offer

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

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

P(J' n M) = P(J'|M) × P(M) = 0.60 × 0.20 = 0.12

P(J' n W) = P(J'|W) × P(W) = 0.90 × 0.10 = 0.09

Total probability that she doesn't get job offer

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

= 0.14 + 0.12 + 0.09 = 0.35

Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation?

This probability = P(S|J')

P(S|J') = P(J' n S) ÷ P(J') = 0.14 ÷ 0.35 = 0.400

i. a moderate recommendation?

P(M|J') = P(J' n M) ÷ P(J') = 0.12 ÷ 0.35 = 0.343

ii. a weak recommendation?

P(W|J') = P(J' n W) ÷ P(J') = 0.09 ÷ 0.35 = 0.257

Hope this Helps!!!