Counting Methods & Basic Probability

1. Counting Methods & Basic Probability Thinking Mathematically by Blitzer Sections 11.1 – 11.4

2. Lottery winners • Weather forecasters • Stock market brokers and investors • Life/health insurance actuaries The mathematics of risk is called probability and it can be useful to help us make decisions and plans.

3. Section 11.1 The FUNDAMENTAL COUNTING PRINCIPLE

4. Example You have 2 pairs of pants and 3 shirts. How many outfits can you make with one pair of pants and one shirt? This is a TREE DIAGRAM.

5. Example You have 2 pairs of pants and 3 shirts. How many outfits can you make with one pair of pants and one shirt? 6 outfits 6 4 5 3 1 2

6. Example How many outfits can you make with 2 pairs of pants, 3 shirts, and 4 hats? 24 outfits

7. Example How many outfits can you make with 2 pairs of pants, 3 shirts, and 4 hats? 24 outfits Do I have to make a tree in order to determine the number of outfits?

8. Example How many outfits can you make with 2 pairs of pants, 3 shirts, and 4 hats? Fundamental Counting Principle If one item from a group of M items is chosen and another item from a group of N items is chosen, the number of two-item choices is MN .

9. Example How many different drinks can you get from the Coke Freestyle machine? M  N M = 22 N = 8

10. Example How many different meals can you create if you choose one appetizer, one main course, and one dessert?

11. Example How many different meals can you create if you choose one appetizer, one main course, and one dessert?

12. Example A multiple choice quiz is given. There are five questions and for each you may choose A, B, C, or D. In how many ways can you answer the quiz? __ __ __ __ __

13. Example A quiz is given. There are 3 multiple choice questions, each with 4 choices. There are 3 true/false questions. In how many ways can you answer the quiz? __ __ __ __ __ __

14. Example Telephone numbers in the U.S. have a 3-digit area code and a 7-digit number. Neither the first digit of the area code nor the the first digit of the number can be 0 or 1. How many possible phone numbers are there? ( __ __ __ ) __ __ __ – __ __ __ __

15. Section 11.2 PERMUTATIONS

16. Example Three students want snack cart. Find the number of ways they can line up. 6 ways

17. Example Three students want snack cart. Find the number of ways they can line up. __ __ __

18. Example Four musical groups play at a concert. Find the number of ways that the performances can be arranged. __ __ __ __

19. Example Four musical groups play at a concert. Find the number of ways that the performances can be arranged. __ __ __ __ Permutation– an ordered arrangement of items in which • no item is used more than once and • the order of the arrangements makes a difference

20. Example In how many ways can 7 books be arranged on a shelf if the order matters to you? __ __ __ __ __ __ __ Factorial notation – if n is a natural number, the notation n! is called “n factorial” and is the product of all the natural numbers from n down to 1.

21. Example Natural numbers are counting numbers: 1, 2, 3, 4, 5, . . . Factorial is not defined for numbers other than these, except: 0! = 1 Factorial notation – if n is a natural number, the notation n! is called “n factorial” and is the product of all the natural numbers from n down to 1.

22. Example How many “words” can be made from the word OSCAR? That is, in how many ways can the letters in the word OSCAR be arranged (and order matters)?

23. Let’s work with factorials. Evaluate:

24. Let’s work with factorials. Evaluate:

25. Let’s work with factorials. Evaluate:

26. Let’s work with factorials. Evaluate:

27. Example There are 15 students in a class. Three students of these 15 may line up to get snack cart. In how many ways can those 3 people line up? This is an ordered arrangement with no item used more than once and order matters… permutation! __ __ __

28. Formula Permutation of n items taken r at a time: and are other notations used for permutations.

29. Example Ten people run in a race in which prizes will be awarded for 1st and 2nd place. In how many ways can 1st and 2nd place be awarded? We need the permutation of 10 items taken 2 at a time.

30. Example A little league baseball team has 13 players. A batting order of 9 players must be chosen. In how many ways can batting order be established? We need the permutation of 13 items taken 9 at a time.

31. Example How many “words” can be made from the word OSCAR ? We need the permutation of 5 things taken 5 at a time. Now let’s work on a couple of special cases with “words.”

32. Example How many “words” can be made from the word MATH ?

33. Example How many “words” can be made from the word ALGEBRA ? Permutation of duplicate items The number of permutations of n items taken r at a time, when q of the items are identical:

34. Example How many “words” can be made from the word EMERGE ?

35. Example How many “words” can be made from the word MISSISSIPPI ?

36. HOMEWORK From the Cow book 11.1 pg564 # 1 – 21 odd 11.2 pg 571 # 1 – 21 odd, 33 – 55 odd