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

ways of getting 8 nonspades from the 39 nonspades in the deck is , and hence, the number of ways five spades and eight nonspades occurs in a hand of 13 cards is the product . This is the number of elements in the sample space constituting the event of “getting five spades in dealing 13 cards from a shuffled deck.” Since the sample space consists of equally likely sample points, each sample point is assigned the same probability . Hence, the probability of getting five spades in dealing 13 cards is

      PRACTICE PROBLEMS FOR SECTION 3.4

      1 In a certain clinical trial, a medical team wants to study four different doses of a new medication for cervical cancer in five patients. In how many different ways can the team select one dose of the medication and one of the patients?

      2 A small motel with nine rooms has three twin beds in two rooms, two twin beds in three rooms, and one twin bed in rest of the four rooms. In how many different ways can the manager of the motel assign these rooms to a group of 16 guests who told the manager that they have no preference about whom they share the room with?

      3 In how many ways can a class of 30 students select a committee from the class that consists of a president, a vice president, a treasurer, and a secretary (a) if any student may serve either of these roles but no student may serve in multiple roles and (b) if any student may serve in multiple roles?

      4 A multiple‐choice board exam consists of 15 questions, each question having four possible answers. In how many different ways can a candidate select one answer to each question?

      5 A chain restaurant offers a dinner menu with four different soups, three different salads, 10 entrees, and four desserts. In how many ways can a customer choose a soup, a salad, an entrée, and a dessert?

      6 If in Problem 3 above, the committee consists of just four members, then in how many ways can the class select the committee?

      7 If 13 cards are dealt from a thoroughly shuffled deck of 52 ordinary playing cards, find the probability of getting five spades and four diamonds.

      8 How many different permutations can be obtained by arranging the letters of the word engineering? Cardiologist?

      9 A cholesterol‐lowering drug is manufactured by four different pharmaceutical companies in five different strengths and two different forms (tablet and capsule). In how many different ways can a physician prescribe this drug to a patient?

      10 How many different car plates can be issued if the Department of Motor Vehicles decides to first use two letters of the English alphabet and then any four of the digits ?

      11 In a random experiment, one die is rolled, one coin is tossed, and a card is drawn from a well‐shuffled regular deck of playing cards and its suit noted. How many sample points are there in the sample space of this random experiment?

      12 Each of 10 websites either contains (C) an ad of a car manufacturer or does not contain the ad (N). How many sample points are there in the sample space of a random experiment that selects a website at random?

3.5 Conditional Probability

      In some probability problems, we are asked to find the probability that an event F occurs when it is known or given that an event E has occurred. This probability, denoted by and called the conditional probability of F given E, is obtained essentially by letting E be a new sample space, sometimes known as an induced sample space, and then computing the fraction of probability on E that lies on , that is,

(3.5.1)

where, of course, . Similarly

      (3.5.2)

      where . The conditional probability of , given that E has occurred, using (3.5.1), is

Since we are in the “new” sample space E, we find that (see Figure 3.6.1)

      That is, the rule of complementation is preserved in the induced sample space E.

      Example 3.5.1 (Concept of conditional probability) The manufacturing department of a company hires technicians who are college graduates as well as technicians who are not college graduates. Under their diversity program, the manager of any given department is careful to hire both male and female technicians. The data in Table 3.5.1 show a classification of all technicians in a selected department by qualification and gender. Suppose that the manager promotes one of the technicians to a supervisory position.

      1 If the promoted technician is a woman, then what is the probability that she is a nongraduate?

      2 Find the probability that the promoted technician is a nongraduate when it is not known that the promoted technician is a woman.

      Solution: Let S be the sample space associated with this problem, and let E and F be the two events defined as follows:

       E: the promoted technician is a nongraduate

       F: the promoted technician is a woman

      In Part (a) we are interested in finding the conditional probability .

Since any of the 100 technicians could be promoted, the sample space S consists of 100 equally likely sample points. The sample points that are favorable to the event E are 65, and those that are favorable to the event F are 44. Also the sample points favorable to both the events E and F are all the women who are nongraduates and equal to 29. To describe this situation, we have

      (a) Therefore,

      and for part (b), we have that .

Table 3.5.1 Classification of technicians by qualification and gender.

	Graduates	Nongraduates	Total
Male