1. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 7.3 Using Counting Methods to Find Probability

2. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Objective o Use the Fundamental Counting Principle to calculate probabilities

3. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Using Counting Methods to Find Probability Our goal is to be able to calculate classical probability for certain events. Now that we’ve looked at several methods of counting the outcomes in a sample space, we can begin to look at calculating probabilities.

4. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Calculating Classical Probability Using Combinations Suppose that as one of the 20 graduate students in the physics department, you have a chance of being selected for one of the three student spots for a conference trip to Cancun. If the names of all the graduate students were put in a hat and three were drawn, what is the probability that you and your two friends, Leonard and Sheldon, end up being chosen?

5. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Calculating Classical Probability Using Combinations (cont.) Solution The first thing we need to do is count the number of ways that the three student spots on the trip can be filled. Because the order in which the students are chosen is not important, we can count the outcomes using the combination formula.

6. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Calculating Classical Probability Using Combinations (cont.) We have 20 students to choose from, so n = 20 and r = 3. That means there are possible ways to choose 3 students from 20 for the trip.

7. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 1: Calculating Classical Probability Using Combinations (cont.) There is only one way in which to choose you and your two friends, so the probability of this event happening is In other words, it is very unlikely that the three of you would randomly be chosen to go on the conference trip to Cancun together.

8. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Classical Probability Using Permutations Let’s change the previous example slightly. Suppose that as one of the 20 graduate students in the physics department, you have a chance of being selected for one of the three student spots for a conference in Cancun. The names of all the graduate students are put in a hat and three are drawn. However, if your name is drawn first, you get all expenses paid. If you are chosen second, everything is paid for except meals, and if you’re chosen third, you must pay for your own meals and hotel.

9. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Classical Probability Using Permutations (cont.) (This means that the department still picks up the tab for the flight and conference fees of the three lucky students, so it’s not a bad deal!) What is the probability that you and your two friends, Leonard and Sheldon, all end up being chosen, and that your name is drawn first?

10. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Classical Probability Using Permutations (cont.) Solution This time, the order in which the three students are chosen does make a difference when we are counting, so we’ll use a permutation. Note that n is still 20 and r is still 3. Now there are possible ways to choose the three lucky students for the trip.

11. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Classical Probability Using Permutations (cont.) However, let’s consider how many outcomes are in the event that you, Sheldon, and Leonard are chosen, and that your name is drawn first; we'll call this event E. Let’s list all the ways that the three of you could be chosen.

12. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Classical Probability Using Permutations (cont.) Table 1 : Combinations of You, Leonard, and Sheldon Possibility 1Possibility 2Possibility 3 1 st pick: You 1 st pick: Sheldon 2 nd pick: Leonard2 nd pick: Sheldon2 nd pick: You 3 rd pick: Sheldon3 rd pick: Leonard Possibility 4Possibility 5Possibility 6 1st pick: Sheldon1 st pick: Leonard 2 nd pick: Leonard2 nd pick: Sheldon2 nd pick: You 3 rd pick: You 3 rd pick: Sheldon

13. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 2: Calculating Classical Probability Using Permutations (cont.) We can see that there are only two ways in which you are first in the list, and therefore get all expenses paid. So the probability that the event described occurs in this way is It seems that there is an even smaller chance of this happening, so it’s better not to be greedy and wish for the top spot!

14. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Complement The complement of event E, denoted by E c, consists of all outcomes in the sample space that are not in event E.

15. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Finding the Complement of an Event Describe the complement for each of the following events. a. Rolling an even number on a die. b. Choosing a number that doesn’t end in 1, from all positive two-digit whole numbers. c. From a class of 52 students, choosing a student who is over 21 years old.

16. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 3: Finding the Complement of an Event (cont.) Solution a. The complement contains all the odd numbers on a die (that is, 1, 3, and 5). b. The set of all positive two-digit whole numbers includes the numbers 10 through 99. The complement of our event would be all two-digit numbers that do end in 1 (that is, 11, 21, 31, 41, 51, 61, 71, 81, and 91). c. The complement consists of the students in the class who are 21 years old or younger.

17. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Skill Check #1 A pair of dice is rolled and the resulting sum is odd. Which of the following outcomes could be in the complement of this event? a. A sum greater than 8 b. A sum that is an even number c. A sum less than 5 d. A sum that is a multiple of 3 e. All of the above Answer: e. all of the above

18. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Complement Rules of Probability

19. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements Using the data given, find the following probabilities involving nuts imported into the United States between 2006-2011.

20. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.) Table 2: US Import Destinations by Weight (in pounds) for Fresh or Dried Walnuts and Pistachios, 2006–2011 Walnuts India4373 Mexico1268 Spain5239 China1533 Austria1938 Other countries4157 Pistachios Iran2012 Turkey2030 Hong Kong262 Switzerland64 Italy115 Other countries323 Source: USDA. “Fruit and Tree Nut Data.” http://www.ers.usda.gov/data- products/fruit-and-tree-nut-data/data-by-commodity.aspx

21. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.) a. The probability that the walnuts you consumed during this period were from Austria. b. The probability that the walnuts you consumed during this period were from somewhere other than Austria. c. Assume that you also purchased pistachios during this time period. What is the probability that the pistachios came from somewhere other than Italy and Switzerland?

22. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.) Solution a. The probability that the walnuts you consumed during this period were from Austria is found by dividing the weight of walnuts imported from Austria by the weight of total walnuts imported during this time period. The first number is given in the table as 1938 pounds.

23. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.) To find the total weight of walnuts imported, we need to add together all of the weights of walnuts imported. The probability that the walnuts were from Austria is then found by

24. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.) b. We could find the probability that the walnuts came from somewhere other than Austria by combining all the remaining places together. However, given that we just calculated the probability that the walnuts were from Austria, it is easier for us to just calculate the complement.

25. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.) c. Again, it will be easier for us to calculate the complement here rather than all the other possibilities. First calculate the probability that the pistachios you consumed came from either Italy or Switzerland.

26. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.) Now, to calculate the probability that the pistachios came from somewhere other than Switzerland or Italy, we’ll find the complement.

27. HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Example 4: Finding Probability Using Complements (cont.)