A worker has asked her supervisor for a letter of...

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, or weak are $.7, .2$, and .1, respectively. (a) How certain is she that she will receive the new job offer? (6) Given that she does receive the offer, how likely should she feel that she received a strong recommendation; a moderate recommendation; a weak recommendation? (c) Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation; a moderate recommendation; a weak recommendation?

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, or weak are $.7, .2$, and .1, respectively.
(a) How certain is she that she will receive the new job offer?
(6) Given that she does receive the offer, how likely should she feel that she received a strong recommendation; a moderate recommendation; a weak recommendation?
(c) Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation; a moderate recommendation; a weak recommendation?

Key Concepts

Conditional Probability

This is the probability of one event occurring given that another event has already occurred. It is essential in decision-making processes where an outcome depends on a certain condition or set of conditions being met.

Total Probability Theorem

This theorem allows the calculation of the overall probability of an event by considering all the possible ways in which that event can occur, each weighted by their respective probabilities. It is particularly useful when the event can be broken down into several mutually exclusive scenarios.

Bayes' Theorem

Bayes' Theorem provides a way to update prior beliefs or probabilities when new evidence is introduced. It relates the conditional and marginal probabilities of events and enables the computation of the probability of a cause given an observed outcome.

Transcript

00:01 Okay, so here we're going to have that mew is equal to $37 ,000, sigma is equal to $4 ,000, and p is equal to $0 .8.

00:08 So in part a, we are considering a random sample of 60 students from this college, and we are interested in computing the probability that less than 70 % of the students from this school have accepted job offers.

00:21 So we're calculating the probability that p hat is going to be less than or equal to 0 .6.

00:31 So we get that mu is equal to p, which is equal to 0 .8.

00:36 Sigma is then equal to the square root of p times 1 minus p all over n.

00:42 That gives us 0 .0516.

00:47 And then the probability here, probability of p hat being less than or equal to 0 .7 is going to be equal to the probability that z is less than or equal to 0 .7, minus 0 .8, all divided by 0 .0 .0516.

01:08 That is going to be equal to the probability that z is less than equal to negative 1 .94, which gives us 0 .062.

01:20 So there we have the probability that less than 70 % of the students from this sample have accepted, the drop offers is going to be 0 .062.

01:30 And then in part b, we consider a random sample of six students from this college.

01:36 So calculating the probability that p hat is less than equal to 0 .7, again, sigma here is going to be equal to the square root of p times 1 minus p over n.

01:45 That's the square root of 0 .8 times 0 .2, all over 6, which gives us 0 .1633.

01:53 And then the probability is going to be equal to the probability that z is less than or equal to 0 .7 minus 0 .8, all divided by 0 .1633, and that is going to be equal to 0 .2709.

02:16 So that's the probability that 70 % of the students from this sample have accepted the job.

02:23 Offers is going to be 0 .2709...

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