SOLVED - how to calculate conditional probability with multiple probabilties

Question

I'm trying to solve the following homework problem:

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 .7, .2, and .1, respectively. How certain is she that she will receive the new job offer?

I'm assuming this involves Bayes Theoerem in some manner: P(A) * P(B|A) = P(B) * P(A|B)

So I suppose P(A) could be P(getting a job), but then how would you fit into the formula the multiple types of recommendation and how that changes the probability? Would P(B) be P(getting a strong recommendation)? But then how do the other types of recommendations fit? Is this not a Bayes Theorem problem at all?

Answer 1

Ignoring moderate, suppose she can either get a strong or a weak letter.

Suppose she gets a strong letter. Then her probability of getting the job is the probability of the job given that she got a strong letter.

$$P(J|S)$$

Conversely, suppose she gets a weak letter. Her probability of getting the job is the probability of getting the job given that she got a weak letter.

$$P(J|W)$$

Say we kind of know these two quantities. We think

$$P(J|S) = 0.8$$

$$P(J|W) = 0.1$$

We also know how likely it is she will get one or the other letters

$$P(S) = 0.7$$

$$P(W) = 0.3$$

(note that $P(S) + P(W) = 1$ because they are the only two options).

We can write

$$P(J) = P(J|S)P(S)+P(J|W)P(W)$$

This is kind of an expression of two different possible paths. In one, we get a strong letter and wonder whether she'll get the job given that strong letter. In the other, we get a weak one and wonder whether she'll get the job given the weak letter. Since the probability that we get a strong letter is so much greater than the probability that we get a weak letter, we $weigh$ the path containing the strong letter more heavily.

This makes use of the law of total probability.