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!!!