BYES' THEOREM



                  
BYES' THEOREM

 It is a conditional law of probability. Two levels of probability are considered here. The statistics is calculated using Byes' theorem. 
  1.      A factory has two machines. Past records show that machine ‘A’ produces 35% of items of the output and m/c ‘B’ produces remaining. Further 6% of items produced by m/c A were defectives and 3% defectives from m/c B. If a defective item is drawn at random, what is the probability that the defective item was produced by m/c A and m/c B?
  2.    Four plants producer an item , the first plant producer 5000 of  units , second 3000 , third 8000 ,  and fourth 4000 of units. The event Ai: a unit is produced in plant i. i = 1,2,3,4 and event B:a unit is defective. From the records of the proportions  of defectives produced at each plant the following  conditional  probabilities are set : P(B/A1)=0.05 , P(B/A2)=0.10 , P(B/A3)=0.15 , P(B/A4)=0.02.A unit of the product made at one of these plants is tested  and  is found to be defective. What is probability that the unit was produced is plant 1, 2, 3 and 4.
  3.   An executive goes to office by car, public Bus or Train. Historical record shows that she has used Car 20% times. The use of Bus and Train was 35% and 45% times. The probability of reaching office before 10.00 AM when she used Car is 0.80. That for Bus and Train is 0.50 and 0.70. What is probability that she will reach office after 10.00 AM on a given day? If she reached office after 10.00 AM then what is probability that she had used Car, Bus or Train?
  4.  An electronic plant has 20% of electronic components stored at humid place. Other 80% are placed at dry place. The probability of failure of the component stored at humid place is 0.60 and that for dry place is 0.20. If a component is selected randomly, then what is probability that it will fail. Also if a component found failed what is probability that it had been stored at damp or dry place?
  5. 90% of the trained employees meet the target. 60% of untrained employee can meet the target. If there are 75% employees trained; what is probability that a randomly selected employee can meet the target?
  6.       In Orange County, 51% of the adults are males. (It doesn't take too much advanced mathematics to deduce that the other 49% are females.) One adult is randomly selected for a survey involving credit card usage.
    a. Find the prior probability that the selected person is a male.
    b. It is later learned that the selected survey subject was smoking a cigar. Also, 9.5% of males smoke cigars, whereas 1.7% of females smoke cigars (based on data from the Substance Abuse and Mental Health Services Administration). Use this additional information to find the probability that the selected subject is a male.
  7. An aircraft emergency locator transmitter (ELT) is a device designed to transmit a signal in the case of a crash. The Altigauge Manufacturing Company makes 80% of the ELTs, the Bryant Company makes 15% of them, and the Chartair Company makes the other 5%. The ELTs made by Altigauge have a 4% rate of defects, the Bryant ELTs have a 6% rate of defects, and the Chartair ELTs have a 9% rate of defects (which helps to explain why Chartair has the lowest market share).
    a. If an ELT is randomly selected from the general population of all ELTs, find the probability that it was made by the Altigauge Manufacturing Company.
    b. If a randomly selected ELT is then tested and is found to be defective, find the probability that it was made by the Altigauge Manufacturing Company.
  8.            Pregnancy Test Results: refer to the results summarized in the table below.
     Situation 
    Positive Test Result (Pregnancy is indicated)
    Negative Test Result (Pregnancy is not indicated)
    Subject is Pregnant
    80
    5
    Subject is Not Pregnant
    3
    11

    a. If one of the 99 test subjects is randomly selected, what is the probability of getting a subject who is pregnant?
    b. A test subject is randomly selected and is given a pregnancy test. What is the probability of getting a subject who is pregnant, given that the test result is positive?
    c. One of the 99 test subjects is randomly selected. What is the probability of getting a subject who is not pregnant?
    d. A test subject is randomly selected and is given a pregnancy test. What is the probability of getting a subject who is not pregnant, given that the test result is negative?
  9. In a study of pleas and prison sentences, it is found that 45% of the subjects studied were sent to prison. Among those sent to prison, 40% chose to plead guilty. Among those not sent to prison, 55% chose to plead guilty.
    a. If one of the study subjects is randomly selected, find the probability of getting someone who was not sent to prison. (Ans:0.55) .
    b. If a study subject is randomly selected and it is then found that the subject entered a guilty plea, find the probability that this person was not sent to prison. (Ans: 0.627)
  10. A diagnostic test has a probability 0.95 of giving a positive result when applied to a person suffering from a certain disease, and a probability 0.10 of giving a (false) positive when applied to a non-sufferer. It is estimated that 0.5 % of the population are sufferers. Suppose that the test is now administered to a person about whom we have no relevant information relating to the disease (apart from the fact that he/she comes from this population). Calculate the following probabilities:
    (a) that the test result will be positive;
    (b) that, given a positive result, the person is a sufferer;
    (c) that, given a negative result, the person is a non-sufferer;
    (d) that the person will be mis-classified.
  11. There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?

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