Math 115 A - Section 3 - Test 1 Name:___________________04 October 2000In order to get full credit, you must show all of the steps and all of the work that youuse to solve each problem.A list of formulas that you may find useful is provided on the last page of this test.Question Points Score1 302 73 154 125 186 18Total 1001. You are applying for jobs and have been interviewed by only two differentcompanies, called Alpha Co and Beta Co respectively. You estimate that there is a60% chance that you will get an offer from Alpha Co and a 45% chance that you willget an offer from Beta Co. You also know that there is a non-zero probability that youwill get no offer at all.Note: the questions (i) to (v) below are independent. If you do not know how toanswer one of them, move on to the next one.(i) (4 points) Set up a sample space for this problem and use the informationgiven above to draw a diagram showing the different events (that you willhave to define). (ii) Assume that there is a 25% chance that you will get an offer from bothcompanies. - (4 points) What is the probability that you will get at least one offer ?Explain.- (4 points) What is the probability that you will get no offer at all ?Explain.(iii) (5 points) Assume that there is a 10% chance that you will get no offer at all.What is the probability that you get both offers ? Justify your answer.(iv) Assume that the events of getting an offer from Alpha Co and getting an offerfrom Beta Co are independent.- (4 points) What is the probability that you will get both offers ? Explain.- (3 points) Explain in real-world terms whether you believe that getting anoffer from Alpha Co and getting an offer from Beta Co are independent. (v) (6 points) Assume that there is a 20% chance that you will get both offers. Ifyou know that you are going to get an offer from Alpha Co, what is theprobability that you will also get an offer from Beta Co ?2. Consider two events R and W. (i) (3 points) Shade the event W C on the diagram below.(ii) (4 points) Give a formula for the event shaded below. Also, describe in wordswhat this event corresponds to.3. Assume that the random variable X can only take values 0, 1, 2 and 3, withprobabilities: P(X=0) = 0.15, P(X=1) = 0.35, and P(X=2) = 0.10.(i) (10 points) Find the expected value of X. Explain what this number means.(ii) (5 points) Find P(X < 4) and P(0 - X - 1).4. (12 points) You are a loan officer and have to decide whether you should foreclose onone of your clients. You are given the following information: the full value of thisR WR Wperson's loan is $700,000; the foreclosure value is $300,000 and the default value is$100,000. You looked at past bank records and you estimated that the probability ofsuccess of an attempted workout is 0.45. Use this information to decide whether yourecommend workout or foreclosure for this loan. Explain your decision.5. You have a coin, which is such that the probability of getting a head is 0.55.(i) (3 points) Is this a fair coin ? Explain.(ii) (15 points) Let X be the number of heads you obtain when you toss the cointhree times. You can assume that the tosses are independent from one another.Find P(X=1). Hint: set up a sample space for the experiment of tossing thecoin three times and identify the events for which X=1. Then, find thecorresponding probabilities.6. You are organizing a conference and know that all of your out-of-town participantswill come by plane. There are three airlines they can use, called Airline One, AirlineTwo and Airline Three. Let A be the event that a randomly selected out-of-townparticipant flies on Airline One, let B be the event that she flies on Airline Two and Cbe the event that she flies on Airline Three. You know that P(A) = 0.45, P(B) = 0.30,and P(C) = 0.25. You also know that a flight arriving to your city is on time with thefollowing probabilities: 80% for Airline One, 75% for Airline Two and 60% forAirline Three. (i) (4 points) Let S be the event that a randomly selected out-of-town participantarrives on time. Describe in words the meaning of P(S | A).(ii) (14 points) Find the probability that a randomly selected out-of-townparticipant arrives on time. Justify your answer.FORMULAS FEPFPEPFEP )()()(1)( EPEPC CCCFEFE CCCFEFE xxXPxXEall)()()B(P)BA(P)B|A(P
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