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
(3.5.1)
where, of course,
(3.5.2)
where
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 | 20 | 36 | 56 |
| Female | 15 | 29 | 44 |
| Total | 35 | 65 | 100 |
Note that the probability P(E), sometimes known as the absolute probability of E, is different from the conditional probability P(E|F). If the conditional probability P(E|F) is the same as the absolute probability P(E), that is, P(E|F) = P(E), then the two events E and F are said to be independent. In this example, the events E and F are not independent.
Definition 3.5.1 Let S be a sample space, and let E and F be any two events in S. The events E and F are called independent if and only if any one of the following is true:
(3.5.3)
(3.5.4)
(3.5.5)
The
