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Probability
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2.3
Counting Techniques
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3. Counting Techniques
When the various outcomes of an experiment are equally likely (the same probability is assigned to each simple event), the task of computing probabilities reduces to counting.
Letting N denote the number of outcomes in a sample space and N(A) represent the number of outcomes contained in an event A,
(2.1)

4. The Product Rule for Ordered Pairs

5. The Product Rule for Ordered Pairs
Proposition

6. Example 2.18
A family has just moved to a new city and requires the services of both an obstetrician and a pediatrician. There are two easily accessible medical clinics, each having two obstetricians and three pediatricians.
The family will obtain maximum health insurance benefits by joining a clinic and selecting both doctors from that clinic. In how many ways can this be done?
Denote the obstetricians by O1, O2, O3, and O4 and the pediatricians by P1, , P6.

7. Example 2.18
cont’d
Then we wish the number of pairs (Oi, Pj) for which Oi and Pj are associated with the same clinic.
Because there are four obstetricians, n1 = 4, and for each there are three choices of pediatrician, so n2 = 3.
Applying the product rule gives N = n1n2 = 12 possible choices.

8. The Product Rule for Ordered Pairs
In many counting and probability problems, a configuration called a tree diagram can be used to represent pictorially all the possibilities.
The tree diagram associated with Example 2.18 appears in Figure 2.7.
Tree diagram for Example 18
Figure 2.7

9. A More General Product Rule

10. A More General Product Rule

11. Example: License Plates
Suppose the state of South Carolina creates 6 digit license plates with three letters and then three numbers. How many license plates can it make?
How many license plates can it make if they do not want to repeat numbers or letters?

12. Permutations and Combinations

13. Permutations and Combinations
Consider a group of n distinct individuals or objects (“distinct” means that there is some characteristic that differentiates any particular individual or object from any other).
How many ways are there to select a subset of size k from the group?
For example, if a Little League team has 15 players on its roster, how many ways are there to select 9 players to form a starting lineup?

14. Permutations and Combinations
Or if a university bookstore sells ten different laptop computers but has room to display only three of them, in how many ways can the three be chosen?
An answer to the general question just posed requires that we distinguish between two cases. In some situations, such as the baseball scenario, the order of selection is important.
For example, Angela being the pitcher and Ben the catcher gives a different lineup from the one in which Angela is catcher and Ben is pitcher.

15. Permutations and Combinations
Often, though, order is not important and one is interested only in which individuals or objects are selected, as would be the case in the laptop display scenario.
Definition

16. Permutations and Combinations
Proposition

17. Permutations and Combinations
Proposition

18. Example
A local college is investigating ways to improve the scheduling of student activities. A fifteen-person committee consisting of five administrators, five faculty members, and five students is being formed. A five-person subcommittee is to be formed from this larger committee. The chair and co-chair of the subcommittee must be administrators, and the remainder will consist of faculty and students. How many different subcommittees could be formed?

19. Example Choose two administrators. Choose three faculty and students.
Two sub-choices:
Choose two administrators.
Choose three faculty and students.
Number of choices:

20. Arrangments
How many “words” can be formed from the word MISSISSIPPI? By words I just mean arrangements of letters…

21. Example 2.22
A particular iPod playlist contains 100 songs, 10 of which are by the Beatles.
Suppose the shuffle feature is used to play the songs in random order (the randomness of the shuffling process is investigated in “Does Your iPod Really Play Favorites?”
What is the probability that the first Beatles song heard is the fifth song played?
In order for this event to occur, it must be the case that the first four songs played are not Beatles’ songs (NBs) and that the fifth song is by the Beatles (B).

22. Example 2.22
cont’d
The number of ways to select the first five songs is 100(99)(98)(97)(96).
The number of ways to select these five songs so that the first four are NBs and the next is a B is 90(89)(88)(87)(10).
The random shuffle assumption implies that any particular set of 5 songs from amongst the 100 has the same chance of being selected as the first five played as does any other set of five songs; each outcome is equally likely.

23. Example 2.22
cont’d
Therefore the desired probability is the ratio of the number of outcomes for which the event of interest occurs to the number of possible outcomes: