Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP. Bhisham C. Gupta

with the following example.

      Example 3.6.2 (Applying Bayes's theorem) David, Kevin, and Anita are three doctors in a clinic. Dr. David sees 40% of the patients, Dr. Anita sees 25% of the patients, and 35% of the patients are seen by Dr. Kevin. Further 10% of Dr. David's patients are on Medicare, while 15% of Dr. Anita's and 20% of Dr. Kevin's patients are on Medicare. It is found that a randomly selected patient is a Medicare patient. Find the probability that he/she is Dr. Kevin's patient.

      Solution: Let

        = Person is Dr. Kevin's patient

        = Person is a Medicare patient

and let

        = Person is Dr. Anita's patient

        = Person is Dr. David's patient

      We are given that , , while , and . We wish to find . Using (3.6.4) with , we have

      We note that the posterior probability of , given E, is 0.475, while the prior probability of was . We sometimes say that the prior information about has been updated in light of the information that E occurred to the posterior probability of , given E, through Bayes's theorem.

      PRACTICE PROBLEMS FOR SECTIONS 3.5 AND 3.6

      1 A regular die is rolled. If the number that showed up is odd, what is the probability that it is 3 or 5?

      2 Three balanced coins are tossed simultaneously. What is the probability that exactly two heads appear given that at least one head has appeared?

      3 Suppose that and are five mutually exclusive and exhaustive events in a sample space S, and suppose that , and . Another event E in S is such that , and . Find the probabilities , and .

      4 Suppose that four attorneys , and deal with all the criminal cases in a district court. The following table gives the percentages of the cases that each of these attorneys handles, and also the probability that each loses the case;AttorneyProbability of handling the caseProbability of losing the case0.400.150.250.300.250.200.100.40Suppose that a criminal case was lost in the court. Find the probability that this case was handled by Attorney .

      5 Suppose that a random experiment consists of randomly selecting one of four coins , and , tossing it and observing whether a head or a tail occurs. Further suppose that the coins , and are biased such that the probabilities of a head occurring for coins , and are 0.9, 0.75, and 0.60, respectively, while the fourth coin is a fair coin.If the outcome of the experiment was a head, find the probability that coin was tossed.If the outcome of the experiment was a tail, find the probability that coin was tossed.

      6 An industry uses three methods, and to manufacture a part. Of all the parts manufactured, 45% are produced by method , 32% by method , and the rest 23% by method . Further it has been noted that 3% of the parts manufactured by method are defective, while 2% manufactured by method and 1.5% by method are defective. A randomly selected part is found to be defective. Find the probability that the part was manufactured by (a) method , (b) method .

      7 There are four roads connecting location A and location B. The probabilities that if a person takes Road I, Road II, Road III, or Road IV from location A to B, then he/she will arrive late because of getting stuck in the traffic are 0.3, 0.20, 0.60, and 0.35, respectively. Suppose that a person chooses a road randomly and he/she arrives late. What is the probability that the person chose to take Road III?

      8 Suppose that in a ball‐bearing manufacturing plant four machines , and manufacture 36%, 25%, 23%, and 16% of the ball bearings, respectively. It is observed that the four machines produce 2%, 2.5%, 2.6%, and 3% defective ball bearings, respectively. If the ball bearings manufactured by these machines are mixed in a well‐mixed lot and then a randomly selected ball bearing is found to be defective, find the probability that the defective ball bearing is manufactured by (a) machine , (b) machine , (c) machine , (d) machine .

      9 An urn contains five coins of which three are fair, one is two‐headed and one is two‐tailed. A coin is drawn at random and tossed twice. If a head appears both times, what is the probability that the coin is two‐headed?

      

3.7 Introducing Random Variables

      Suppose that a finite sample space S consists of m elements . There are possible events that can be formed from these elements, provided that the empty event and the entire sample space S are counted as two of the events. This is revealed by the fact that we have the choice of selecting or not selecting each of the m elements in making up an event. Rarely, if ever, is one interested in all these events and their probabilities. Rather, the interest lies in a relatively small number of events produced by specified values of some function defined over the elements of a sample space. For instance, in the sample space S of the possible hands of 13 bridge cards, we are usually interested in events such as getting two aces, or eight spades, or 10 honor cards, and so on.

A real and single‐valued function defined on each element e in the sample space S is called a random variable. Suppose that can take on the values . Let be the events that are mutually exclusive and exhaustive in the sample space S, for which , respectively. Let . Then, we say that is a random variable defined over the sample space S and is a discrete random variable which takes the values with