Probability practice solutions

A pharmaceutical firm has discovered a new diagnostic

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A pharmaceutical firm has discovered a new diagnostic test for a certain disease that has  infected 1% of the population. The firm has announced that 95% of those infected will  show a positive test result, while 98% of those not infected will show a negative test  result. What proportion of test results are correct? ANSWER:   Let I = Infected , P = Positive test result      P(Correct)  = P(I and P) + P(I C  and P C ) = P(I)P(P|I) +P(I C )P(P C |I C ) =(.01)(.95) + (1-.01)(.98) = 0.9797 A Ph.D. graduate has applied for a job with two universities:  A  and  B . The graduate feels  that she has a 60% chance of receiving an offer from university  A  and a 50% chance of  receiving an offer from university  B . If she receives an offer from university  B , she believes  that she has an 80% chance of receiving an offer from university  A . Write down what is given: P(A) = .60, P(B) = .50, P(A|B) = .80 25. What is the probability that both universities will make her an offer?  P(A and B) = P(A|B)P(B) = (.80)(.50) = 0.40 4
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26. What is the probability that at least one university will make her an offer?  P(A or B) = P(A) + P(B) – P(A and B) = .60 + .50 - .40 = .70 27. If she receives an offer from university  B , what is the probability that she will not receive an  offer from university  A P(A C |B) = 1 – P(A|B) = 1 - .80 = .20 28. At the beginning of each year, an investment newsletter predicts whether or not the stock  market will rise over the coming year. Historical evidence reveals that there is a 75% chance that  the stock market will rise in any given year. The newsletter has predicted a rise for 80% of the  years when the market actually rose, and has predicted a rise for 40% of the years when the  market fell. Find the probability that the newsletter’s prediction for next year will be correct. ANSWER:   Let R = Rise, PR = Predicted a Rise  P(Correct) = P(R and PR) + P(R C  and PR C ) =P(R)P(PR|R) + P(R C )P(PR|R C ) =(.75)(.80) + (.25)(.60)  =.75 A standard admissions test was given at three locations. One thousand students took the test  at location  A , 600 students at location  B , and 400 students at location  C . The percentages of  students from locations   A ,   B , and   C , who passed the test were 70%, 68%, and 77%,  respectively. One student is selected at random from among those who took the test.  Givens: P(A) = 1000/2000 = .5, P(B) = 600/2000 = .30, P(C) = 400/2000 = .2,  P(Pass|A) = .70, P(Pass|B) = .68, P(Pass|C) = .77 29. What is the probability that the selected student passed the test?  Answer: P(Pass) = P(Pass and A) + P(Pass and B) + P(Pass and C)      [law of Tot. Prob]
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