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Pat has asked a professor for a recommendation letter for graduate school. She estimates that the probability that the letter will be strong is 0.5 and that the probability that the letter will be weak is 0.2, and mediocre 0.3. Pat also estimates that if the letter is strong, the probability that she will enter grad school is 0.8; if it is weak, 0.1; and if it is mediocre, 0.4. Given that she did enter grad school, what is the probability that the recommendation letter was strong?

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Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

Related questions

An Engineering firm advertises a job offer for recent graduates of Mechanical Engineering through three media: El Heraldo, El Portal de Uninorte graduates and through social networks. It is known that each of these media is consulted by recent graduates of Mechanical Engineering with a probability of 0.1, 0.4 and 0.5 respectively. Likewise, it is known that a recent graduate of Mechanical Engineering applies to a job offer having consulted the Heraldo with a probability of 0.002; the portal of graduates of Uninorte with probability 0.001 and social networks with 0.005. Assume that a recent graduate of Mechanical Engineering consults only one medium.

a) What is the probability that it will be applied to a job offer?

b) If the Engineering firm receives an application to the job offer, what is the probability that the applicant recently graduated from Mechanical Engineering has consulted the Herald?

1. If P(A)= 0.5, and P(B)=0.2, then P(A|B) = 

A.  0.7 

B. 0.3

C. 0.1

D. Cannot be determined from the given information 

2. denotes the number of parts available for assembly at a given station on an assembly line; denotes the number of parts available at another station on the same assembly line. The joint pmf of  and   is depicted below:

                                                                                                                                  X2

    0 1 2 3
  0 0.08 0.07 0.04 0.00
  1 0.06 0.15 0.05 0.04
X1 2 0.05 0.04 0.10 0.06
  3 0.00 0.03 0.04 0.07
  4 0.00 0.01 0.05 0.06

Find

(a)     

(b)  

(c)  The probability that there are twice as many parts waiting at one station than at the other.

 

 

 

Happy Clown Carnival Company is scheduled to appear in a city on a particular Saturday. The profits that will be obtained depend heavily upon the weather. If the weather is rainy, the carnival loses $15,000; if it is not rainy, the carnival makes a profit of $5,000. Historically, the probability of rain on any given day is 10%.

The carnival has to set up equipment for its show, but it can cancel the show prior to setting up its equipment. Cancelling the show results in a loss of $1,000, regardless of the weather. However, for an additional $250 cost, the carnival can postpone its setup decision until the Friday before the scheduled performance. At that time, the carnival can obtain the local weather forecast for Saturday’s weather.

The Weather Bureau has compiled data on its past forecasts. When it was rainy, the Weather Bureau had predicted rain 74% of the time. When it was not rainy, the Weather Bureau had predicted no rain 86% of the time.

(a) Set up the decision tree and determine the decision (SETUP or CANCEL) that maximizes the expected profit without waiting for a weather forecast on the blank page provided. Leave enough space to add to this tree in part (d).(15 pts for correct tree, 12 pts for decision and backup calculations.)

(b) Determine the Expected Value of Perfect Information (EVPI) for the situation depicted in your decision tree from (a).(7 pts.)

(c) Append to your decision tree in (a) a decision tree diagram that includes waiting for the results of the weather forecast. Be sure to clearly label the branches and nodes. (15 pts.)

(d) Determine the two posterior probability distributions for Saturday’s actual weather given the weather forecasts on Friday and put your results in the appropriate places on your tree in (c). (Hint: Bayes. ). (15 pts.)

(e) . Calculate the Expected Values at each appropriate node for the part of the tree diagram you added in (c). (30 pts.)

(f) Based on your results in (e), what is your optimal decision strategy? (5 pts.)

(g) Calculate the Expected Value of Sample Information (EVSI) and the efficiency of sample information for this situation. Also calculate the Expected Net Gain of Sample Information. (ENGSI is the amount you expect to gain from the sample information after accounting for what it cost you to get that information.) (5 pts.) Use this page for your decision tree. Please write clearly enough so I can tell what your answers are.

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