1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 4.5 Combining Probability and Counting Techniques With some added content by D.R.S., University of Cordele

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.34: Calculating Probability Using Combinations A group of 12 tourists is visiting London. At one particular museum, a discounted admission is given to groups of at least ten. a.How many combinations of tourists can be made for the museum visit so that the group receives the discounted rate? b.Suppose that a group of the tourists does get the discount. What’s the probability that it was made up of 11 of the tourists?

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.34: Calculating Probability Using Combinations (cont.) a. “How many combinations …?” a.The key words for this problem are at least. If at least 10 are required, then the group will get the discount rate if 10, 11, or 12 tourists go to the museum. We must then calculate the number of combinations for each of the three possibilities. Note that we are counting combinations, not permutations, because the order of tourists chosen to go to the museum is irrelevant.

4. Tourists group discount, part (a) Ways to take 12 tourists 10 at a time? 12 C 10 Ways to take 12 tourists 11 at a time? 12 C 11 Ways to take 12 tourists 12 at a time? 12 C 12 But of course that’s “1”. Use TI-84 nCr to find these values, not the primitive formula.

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.34: Calculating Probability Using Combinations (cont.) As we noted, the question implies that the group will get the discount rate if 10, 11, or 12 tourists attend. Remember, the word “or” tells us to add the results together. So, to find the total number of groups, we add the numbers of combinations together. 66 + 12 + 1 = 79 There are 79 different groups that can be formed to tour the museum at the discounted rate.

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.34: Calculating Probability Using Combinations (cont.) b.the probability that it was made up of 11 of the tourists? Think of the sentence as reading, “Given that a group of tourists received the discount, what’s the probability that it was a group of 11?” So, to find the probability, we need the total number of ways a group of 11 can be chosen from the group of 12 divided by the number of ways that a group could receive the discount. Since both of these were calculated in part a., we can simply substitute these numbers in our fraction.

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.34: Calculating Probability Using Combinations (cont.) Therefore, we calculate the probability as follows. Because there are 12 diff groups of eleven tourists. Because there are 79 diff groups qualify for the discount.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.35: Calculating Probability Using Permutations Jack is setting a password on his computer. He is told that his password must contain at least three, but no more than five, characters. He may use either letters or numbers (0–9). a.How many different possibilities are there for his password if each character can only be used once? b.Suppose that Jack’s computer randomly sets his password using all of the restrictions given above. What is the probability that this password would contain only the letters in his name?

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.35: Calculating Probability Using Permutations (cont.) Solution – How many passwords can be formed? a.First, notice the key words at least and no more than. These words tell us that Jack’s password can be 3, 4, or 5 characters in length. Also note that for a password, the order of characters is important, so we are counting permutations. There are 26 letters and 10 digits to choose from, so we must calculate the number of permutations possible from taking groups of 3, 4, or 5 characters out of 36 possibilities.

10. Password Permutations, continued It’s a lot like the tourists. How many ways to take 36 characters 3 at a time? How many ways to take 36 characters 4 at a time? How many ways to take 36 characters 5 at a time? How many ways in total to get 3, 4, or 5 characters?

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.35: Calculating Probability Using Permutations (cont.) To find the total number of possible passwords, once again we need to add all three of the above numbers of permutations together since the solution contains an “or” statement. Therefore, we obtain 46,695,600 passwords. (Imagine how many possibilities there would be if he could use up to eight or nine characters, and mixed upper/lowercase and punctuation characters too!)

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.35: Calculating Probability Using Permutations (cont.) Using only “J”, “A”, “C”, and “K”: b.To find the probability that a randomly chosen password would include only the four letters from Jack’s name, we need the total number of possible passwords calculated in part a. as well as the number of possible passwords made from the four letters in Jack’s name. (That is, the classic fraction, n(outcomes for this event) / n(Sample space) ). The numerator is the number of permutations of 4 things from a set of 4 (ways to arrange J, A, C, K)

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.35: Calculating Probability Using Permutations (cont.) So, the probability is calculated as follows.

14. More with JACK What if the question were “What is the probability that the password will contain all four letters of his name?” What’s “new” here, besides the passwords we counted in the previous problem? How to count the number of passwords that meet the requirements? What is the probability now?

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.36: Using Numbers of Combinations with the Fundamental Counting Principle Tina is packing her suitcase to go on a weekend trip. She wants to pack 3 shirts, 2 pairs of pants, and 2 pairs of shoes. She has 9 shirts, 5 pairs of pants, and 4 pairs of shoes to choose from. How many ways can Tina pack her suitcase? (We will assume that everything matches.)

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.36: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) Solution Begin by thinking of this problem as a Fundamental Counting Principle problem. Consider the act of packing the suitcase to be a multistage experiment. There are three stages—packing the shirts, packing the pants, and packing the shoes. Thus, there are three suitcase “slots” for Tina to fill—a slot for shirts, a slot for pants, and a slot for shoes. To fill these slots, the combination formula must be used because we are choosing some items out of her closet, and the order of the choosing does not matter. Let’s first calculate these individual numbers of combinations.

17. Suitcase, continued Permutations or Combinations? ______ shirts, taken _____ at a time: __________ ______ pants, taken _____ at a time: __________ ______ pair of shoes, taken _____ at a time: __________

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.36: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) The final step in this problem is different than the final step in the last two examples, because of the word “and.” The Fundamental Counting Principle tells us we have to multiply, rather than add, the numbers of combinations together to get the final result. There are then different ways that Tina can pack for the weekend.

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle An elementary school principal is putting together a committee of 6 teachers to head up the spring festival. There are 8 first-grade, 9 second-grade, and 7 third ‑ grade teachers at the school. a.In how many ways can the committee be formed? b.In how many ways can the committee be formed if there must be 2 teachers chosen from each grade? c. Suppose the committee is chosen at random and with no restrictions. What is the probability that 2 teachers from each grade are represented?

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) Solution In how many ways can the committee be formed? a.There are no stipulations as to what grades the teachers are from and the order in which they are chosen does not matter, so we simply need to calculate the number of combinations by choosing 6 committee members from the 8 + 9 + 7 = 24 teachers.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.)

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) b. How many ways if we get two from each grade? For this problem we need to combine the Fundamental Counting Principle and the combination formula. First, we see that there are three slots to fill: one first-grade slot, one second- grade slot, and one third-grade slot. Each slot is made up of 2 teachers, and we must use the combination formula to determine how many ways these slots can be filled.

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) Number of first-grade combinations:

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) Number of second-grade combinations:

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) Number of third-grade combinations:

26. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) Finally, the Fundamental Counting Principle says that we have to multiply together the outcomes for all three slots in order to obtain the total number of ways to form the committee. Thus, there are ways to choose the committee.

27. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) c. P(a committee does have two from each grade) The final part of this question asks us to find the probability that if a committee was chosen at random, it would meet the requirements given in part b. That is, each grade would have two teachers on the committee. We can combine our answers from parts a. and b. to find the answer to this probability question.

28. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4.37: Using Numbers of Combinations with the Fundamental Counting Principle (cont.) In other words, if the members of the committee are randomly selected, there is a 15.73% chance that the committee will be made up of 2 teachers from each grade.

29. Menu planning example 150 guests are coming. Select 5 appetizers, 4 entrees, 3 side dishes. Available 18 appetizers, 9 entrees, 12 side dishes. How many different menu plans are there?

30. Menu planning example Fundamental counting principle: How many ways to choose appetizers, times how many ways to choose entrees, times how many ways to choose side dishes What about the 150 guests?

31. Reading list example There are 10 novels, 8 non-fiction, and 6 self-help books on the shelf. Choose 5 books to take on vacation. How many different book choices?

32. Reading List Example Key insight – in this case, we’re just selecting books It doesn’t matter how many books of which types

33. Reading list example, modified There are 10 novels, 8 non-fiction, and 6 self-help books on the shelf. Choose 5 books to take on vacation, and at least one book must be a self-help book. How many different book choices?

34. Reading list example, modified There are 10 novels, 8 non-fiction,18 books besides the self-help books, and 6 self-help books on the shelf. Choose 5 books to take on vacation, and at least one book must be a self-help book. How many ways to choose 1 self-help + 4 others + How many ways to choose 2 self-help + 3 others + How many ways to choose 3 self-help + 2 others + How many ways to choose 4 self-help + 1 other + How many ways to choose 5 self-help + 0 others

35. Reading list example, modified There are 10 novels, 8 non-fiction,18 books besides the self-help books, and 6 self-help books on the shelf. Choose 5 books to take on vacation, and at least one book must be a self-help book. And… what if the order of the five books matters?

36. Reading list example, modified There are 10 novels, 8 non-fiction,18 books besides the self-help books, and 6 self-help books on the shelf. Choose 5 books to take on vacation, and at least one book must be a self-help book. And… what if the order of the five books matters? > Multiply the number of ways to choose the books by 5!, since each of those ways has 5! possible orders.

37. Mixture of events of different natures How many outcomes for the activity of Tossing a coin 5 times Drawing 2 cards without replacement

38. Mixing events of different natures Fundamental Counting Principle, so multiply The coin tosses are independent events The card drawing is a combination

39. Tossing a coin 10 times … and the number of heads is between 5 and 8

40. Tossing a coin 10 times … and the number of heads is between 5 and 8 10 coin tosses … the sample space has 2 10 outcomes. How many ways to get 5 heads? 10 C 5 How many ways to get 6 heads? 10 C 6 How many ways to get 7 heads? 10 C 7 How many ways to get 8 heads? 10 C 8