APPENDIX C.2 Probability Computations A7 C.2 Probability Computations ■ Find the probability of mutually exclusive events. ■ Find the probability of the complement of an event. ■ Find the odds of an event.

Mutually Exclusive Events In Appendix C.1, you looked at a formula for calculating simple probabilities. In this section, you will study formulas for calculating probabilities of more involved events and combinations of events. First, you will study a formula for the probability of the union of two disjoint (or mutually exclusive) events. Two events A and B (from the same sample space) are mutually exclusive if A and B have no outcomes in common. That is,A ʝ B ϭл. For instance, if two dice are tossed, the event A of rolling a total of 6 and the event B of rolling a total of 9 are mutually exclusive because the following sets have no elements in com- mon. (The ordered pairs represent the numbers on the dice.) A ϭ ͭ͑1, 5͒, ͑2, 4͒, ͑3, 3͒, ͑4, 2͒, ͑5, 1͒ͮ B ϭ ͭ͑3, 6͒, ͑4, 5͒, ͑5, 4͒, ͑6, 3͒ͮ To find the probability that one or the other of two mutually exclusive events will occur, you can add their individual probabilities, as indicated in Figure C.4.

B

A

P(A ∪ B) = P(A) + P(B) FIGURE C.4

Probability of Mutually Exclusive Events If A and B are mutually exclusive events, then for a given experiment the probability that A or B will occur is P͑A ʜ B͒ ϭ P͑A͒ ϩ P͑B͒. A8 APPENDIX C Probability and Probability Distributions

Example 1 Finding the Probability of Mutually Exclusive Events

A card is drawn from a standard deck of 52 cards. What is the probability that it is a (event A ) or a 4 (event B )?

SOLUTION The two events are mutually exclusive because no one card can be both a face card and a 4. Because n͑A͒ 12 3 P͑A͒ ϭ ϭ ϭ n͑S͒ 52 13 and n͑B͒ 4 1 P͑B͒ ϭ ϭ ϭ n͑S͒ 52 13 it follows that P͑A ʜ B͒ ϭ P͑A͒ ϩ P͑B͒ 3 1 4 ϭ ϩ ϭ . 13 13 13 So, you can conclude that the probability of selecting a face card or a 4 is 4 Ϸ 13 0.308.

✓CHECKPOINT 1 A card is drawn from a standard deck of 52 cards. What is the probability that the card is a or an ? ■

Three events A, B, and C in the same sample space are mutually exclusive if A ʝ B ϭл, A ʝ C ϭл, B ʝ C ϭл. The probability that A, B, or C will occur is given by P͑A ʜ B ʜ C͒ ϭ P͑A͒ ϩ P͑B͒ ϩ P͑C͒. This pattern can be applied to four or more events that are mutually exclu-

sive. That is, if A1, A2, A3, . . . , An are mutually exclusive events in the same sample space then ͑ ʜ ʜ ʜ . . . ʜ ͒ ϭ ͑ ͒ ϩ ͑ ͒ ϩ ͑ ͒ ϩ . . . ϩ ͑ ͒ P A1 A2 A3 An P A1 P A2 P A3 P An . APPENDIX C.2 Probability Computations A9

Example 2 Finding the Probability of Mutually Exclusive Events

The weights of adult Lhasa Apsos at a breeding facility are shown in the table.

Weight (in pounds) 6–8 9–11 12–14 15–17 18–20 21–23 Number of dogs 0 2 7 5 3 1

What is the probability that an adult Lhasa Apso chosen at random will weigh between 12 and 20 pounds?

SOLUTION Let A, B,and C represent the following events. A ϭ weight between 12 and 14 pounds B ϭ weight between 15 and 17 pounds C ϭ weight between 18 and 20 pounds Because these three events are mutually exclusive, the probability that the adult Lhasa Apso’s weight will fall between 12 and 20 pounds is P͑A ʜ B ʜ C͒ ϭ P͑A͒ ϩ P͑B͒ ϩ P͑C͒ 7 5 3 ϭ ϩ ϩ 18 18 18 15 ϭ 18 Ϸ 0.833

✓CHECKPOINT 2 In Example 2, find the probability that an adult Lhasa Apso chosen at random weighs between 9 and 17 pounds. ■

If a sample space S is subdivided into a collection of mutually exclusive events A1, A2, A3, . . . , An such that ʜ ʜ ʜ . . . ʜ ϭ A1 A2 A3 An S then the sum of the probabilities of the events must be 1. This Addition Principle is summarized as follows.

Addition Principle

If A1, A2, A3, . . . , An are mutually exclusive events in a sample space S such that ʜ ʜ ʜ . . . ʜ ϭ A1 A2 A3 An S, then ͑ ͒ ϩ ͑ ͒ ϩ ͑ ͒ ϩ . . . ϩ ͑ ͒ ϭ P A1 P A2 P A3 P An 1. A10 APPENDIX C Probability and Probability Distributions

Example 3 Applying the Addition Principle

Two six-sided dice are tossed. Construct a table showing the probabilities for the various totals of the two dice. Then show that the sum of these probabilities is 1.

SOLUTION The outcomes representing various totals are shown in the following table, which is called a frequency distribution. Number of Total Outcomes Outcomes 21͑1, 1͒ 32͑1, 2͒, ͑2, 1͒ 43͑1, 3͒, ͑2, 2͒, ͑3, 1͒ 54͑1, 4͒, ͑2, 3͒, ͑3, 2͒, ͑4, 1͒ 65͑1, 5͒, ͑2, 4͒, ͑3, 3͒, ͑4, 2͒, ͑5, 1͒ 76͑1, 6͒, ͑2, 5͒, ͑3, 4͒, ͑4, 3͒, ͑5, 2͒, ͑6, 1͒ 85͑2, 6͒, ͑3, 5͒, ͑4, 4͒, ͑5, 3͒, ͑6, 2͒ 94͑3, 6͒, ͑4, 5͒, ͑5, 4͒, ͑6, 3͒ 10͑4, 6͒, ͑5, 5͒, ͑6, 4͒ 3 11͑5, 6͒, ͑6, 5͒ 2 12͑6, 6͒ 1

From this frequency distribution, you can construct the following table showing the probability of each of the various totals. This table is called a probability distribution.

Total 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 5 4 3 2 1 Probability 36 36 36 36 36 36 36 36 36 36 36 ✓ CHECKPOINT 3 Note that the events in this probability distribution are mutually exclusive and Five are tossed. Construct a represent every possible total (or outcome). From the Addition Principle, you can table showing the probabilities for see that the sum of the given probabilities is 1. the number of heads that turn up. 1 2 3 4 5 6 5 4 3 2 1 ϩ ϩ ϩ ϩ ϩ ϩ ϩ ϩ ϩ ϩ ϭ 1 Then show that the sum of the 36 36 36 36 36 36 36 36 36 36 36 probabilities is 1. ■

Be sure you see that the probability of A ʜ B is equal to the sum of the prob- abilities of A and B only if A and B are mutually exclusive. For instance,the probability of drawing a heart (event A ) or a face card (event B ) from a standard deck of 52 cards is not equal to P͑A͒ ϩ P͑B͒ because A and B are not mutually A ∩ B exclusive. To find the probability of A ʜ B, where A and B are not necessarily mutually A B exclusive events, use the fact that n͑A ʜ B͒ ϭ n͑A͒ ϩ n͑B͒ Ϫ n͑A ʝ B͒. A From this, it follows that ͑ ʜ ͒ ϭ ͑ ϩ ͑ ͒ Ϫ ͑ ʝ ͒ P(A ∪ B) = P(A) + P(B) − P(A ∩ B) P A B P A) P B P A B FIGURE C.5 as summarized in the following box. (See Figure C.5) APPENDIX C.2 Probability Computations A11

Probability of the Union of Two Events If A and B are events in the same sample space S, then the probability of A ʜ B is P͑A ʜ B͒ ϭ P͑A͒ ϩ P͑B͒ Ϫ P͑A ʝ B͒.

Example 4 Finding the Probability of the Union of Two Events

One card is selected from a standard deck of 52 playing cards. What is the probability that the card is either a heart or a face card.

SOLUTION Because the deck has 13 , the probability of selecting a heart Hearts (event A) is

♥ ∩ 13 1 A 2♥ n(A B) = 3 P͑A͒ ϭ ϭ . Selecting a heart 3♥ 52 4 4♥ 5♥ 6♥ ♥ 7♥ J J♠ Because the deck has 12 face cards, the probability of selecting a face card (event ♥ J♦ 8♥ 9♥ Q B) is K♥ Q♠ ♣ 10♥ ♦ J Q 12 3 K♠ K♦ P͑B͒ ϭ ϭ . Selecting a face card Q♣ 52 13 K♣ Face cards Because three of the cards are hearts and face cards (See Figure C.6) it follows FIGURE C.6 that 3 P͑A ʝ B͒ ϭ . 52

✓CHECKPOINT 4 Finally, applying the formula for the probability of the union of two events, you can conclude that the probability of selecting a heart or a face card is One card is selected from a stan- dard deck of 52 playing cards. P͑A ʜ B͒ ϭ P͑A͒ ϩ P͑〉͒ Ϫ P͑A ʝ B͒ What is the probability that the 13 12 3 22 card is either a or a face ϭ ϩ Ϫ ϭ Ϸ 0.423 52 52 52 52 card? ■

The Complement of an Event The complement of an event A is the collection of all outcomes in the sample space that are not in A. The complement of event A is denoted by AЈ. Because P͑A ʜ AЈ͒ ϭ 1 and because A and AЈ are mutually exclusive, it follows that P͑A͒ ϩ P͑AЈ͒ ϭ 1. So, the probability of AЈ is P͑AЈ͒ ϭ 1 Ϫ P͑A͒.

Probability of the Complement of an Event If A is an event, then the probability of the complement of A is given by P͑AЈ͒ ϭ 1 Ϫ P͑A͒. A12 APPENDIX C Probability and Probability Distributions

For instance, if the probability of winning a game is 1 1 3 P͑A͒ ϭ , then the probability of losing the game is P͑AЈ͒ ϭ 1 Ϫ ϭ . 4 4 4

Example 5 Finding the Probability of the Complement of an Event

The age distribution for the population of the United States (in 2005) is shown in the table. (Source: U.S. Census Bureau)

Age Bracket Number of People Age Bracket Number of People Under 5 years 20,304,000 40–44 years 22,861,000 5–13 years 36,087,000 45–49 years 22,485,000 14–17 years 17,079,000 50–54 years 19,998,000 18–24 years 29,307,000 55–64 years 30,356,000 25–29 years 20,066,000 65–74 years 18,640,000 30–34 years 20,077,000 75 years and 18,150,000 older 35–39 years 21,002,000 Total 296,412,000

If a person is chosen at random from this population, what is the probability that the person is (a) under 18? (b) 18 or older?

SOLUTION a. The number of people who are under 18 (event A ) is n͑A͒ ϭ 20,304,000 ϩ 36,087,000 ϩ 17,079,000 ϭ 73,470,000 So, the probability of choosing a person who is under 18 is 73,470,000 P͑A͒ ϭ Ϸ 0.248. 296,412,000

✓CHECKPOINT 5 b. Because the event corresponding to choosing a person who is 18 or older is the complement of A, it follows that the probability of choosing a person who is In Example 5, what is the probabil- 18 or older is ity that the person is (a) under 25? ͑ Ј͒ ϭ Ϫ ͑ ͒ Ϸ Ϫ ϭ (b) 25 or older? ■ P A 1 P A 1 0.248 0.752.

In Example 5, note the convenience of using the formula for the complement of an event. Without this formula, you would have to determine the number of outcomes in event AЈ by adding many terms. If A and B are any two events in the same sample space, then it must be true that ͑A ʝ B͒ ʝ ͑AЈ ʝ B͒ ϭл. APPENDIX C.2 Probability Computations A13

So ͑A ʝ B͒ and ͑AЈ ʝ B͒ are mutually exclusive. Moreover, because A B ͑A ʝ B͒ ʜ ͑AЈ ʝ B͒ ϭ B it follows that A P͑A ʝ B͒ ϩ P͑AЈ ʝ B͒ ϭ P͑B͒. (See Figure C.7.) A ∩ BA′ ∩ B P(A ∩ B) + P(A′ ∩ B) = P(B) Example 6 Finding the Probability of (ABЈ ʝ ) FIGURE C.7 A college has an undergraduate enrollment of 3000 students. Of these, 860 are business majors (event A ) and 1400 are women (event B ). Of the business majors, 375 are women. The college newspaper staff is conducting a poll and selects undergraduate students at random to answer a survey. a. What is the probability a student who is selected to participate in the poll is a woman and a business major? b. What is the probability a student who is selected to participate in the poll is a woman and not a business major?

SOLUTION a. The probability that a student who is selected is a woman and a business major is 375 1 P͑A ʝ B͒ ϭ ϭ . 3000 8 ✓CHECKPOINT 6 b. Because the probability that a student who is selected is a woman is In Example 6, what is the P͑B͒ ϭ 1400͞3000 ϭ 7͞15, the probability that the student is a woman and probability a student is selected to not a business major is participate in the poll is (a) a man 7 1 41 and a business major? (b) a man P͑AЈ ʝ B͒ ϭ P͑B͒ Ϫ P͑A ʝ B͒ ϭ Ϫ ϭ Ϸ 0.342. and not a business major? ■ 15 8 120

Odds Probabilities are sometimes given in terms of odds, which is the ratio of P͑A͒ to P͑AЈ͒. For instance, the odds that a horse will win a race might be stated as 2 to 1.

Odds If A is an event, then the odds that A will occur are given by P͑A͒ , P͑AЈ͒  0. P͑AЈ͒ Odds are usually read as “P͑A͒ to P͑AЈ͒. ” A14 APPENDIX C Probability and Probability Distributions

Example 7 Finding the Odds That an Event Will Occur

Two six-sided dice are tossed. What are the odds that the total of the two dice will be 7?

SOLUTION From Example 3, you know that the probability that the total will be 7 (event A ) is 6 1 P͑A͒ ϭ ϭ . 36 6 So, the probability that the total will be a number other than 7 is 1 5 P͑AЈ͒ ϭ 1 Ϫ ϭ . 6 6 The odds that the total will be 7 are given by P͑A͒ 1͞6 1 odds ϭ ϭ ϭ P͑AЈ͒ 5͞6 5 which means that the odds are 1 to 5.

✓CHECKPOINT 7 Two six-sided dice are tossed. What are the odds that the total of the two dice will be 8? ■

If you are given the odds that an event A will occur, then you can determine the probability that A will occur by the following formula.

Finding Probability from the Odds If the odds that an event A will occur are n to m, then the probability that A will occur is n P͑A͒ ϭ . m ϩ n Similarly, the probability that A will not occur is m P͑AЈ͒ ϭ . m ϩ n APPENDIX C.2 Probability Computations A15

Example 8 Finding the Probability from the Odds

The odds of a horse winning a race are given as 2 to 1. Assuming these odds to be correct, what is the probability of the horse winning? What is the probability of the horse losing?

SOLUTION Because the odds are n ϭ 2 to m ϭ 1, the probability of the horse winning is n 2 2 P͑A͒ ϭ ϭ ϭ m ϩ n 1 ϩ 2 3 and the probability of the horse losing is m 1 1 P͑AЈ͒ ϭ ϭ ϭ . m ϩ n 1 ϩ 2 3

✓CHECKPOINT 8 In Example 8, suppose the odds of the horse winning the race are 3 to 1. What is the probability of the horse winning? losing? ■

CONCEPT CHECK

1. The ______of an event A is the collection of all outcomes in the sample space that are not in A. PͧAͨ 2. If A is an event, then the ______that A will occur are given by , PͧAЈͨ where PͧAЈͨ  0. 3. Describe when two events A and B are mutually exclusive. 4. Describe the difference between the event A and AЈ. A16 APPENDIX C Probability and Probability Distributions

The following warm-up exercises involve skills that were covered in earlier sections. You will use Skills Review C.2 these skills in the exercise set for this section. For additional help, review Appendix C.1.

In Exercises 1–2, determine the sample space for the given experiment. 1. A letter from the word probability is selected. 2. A card is selected from the set of black cards from a deck of playing cards.

In Exercises 3–4, two six-sided dice are tossed. Find the indicated probability. 3. The probability that the sum is 5. 4. The probability that the sum is less than 9.

Exercises C.2 See www.CalcChat.com for worked-out solutions to odd-numbered exercises.

In Exercises 1–4, a card is drawn from a standard deck 14. Blood Type In Exercise 13, what is the probability that of 52 cards. the patient chosen has type A, type B, or type AB blood? 1. What is the probability that the card is a face card or a 7? 15. Promotional Contest The number of customers who 2. What is the probability that the card is a or an ace? entered a week-long contest is given in the following table. What is the probability that the winner of the contest 3. What is the probability that the card is a club or a diamond? entered on Sunday or Wednesday? 4. What is the probability that the card is a 2, 3, or 7? Sunday 24 Thursday 61 In Exercises 5–10, two six-sided dice are tossed. Monday 32 Friday 32 5. What is the probability that the sum of the two numbers is 3 or 4? Tuesday 29 Saturday 32 6. What is the probability that the sum of the two numbers is Wednesday 42 5 or 7? 7. What is the probability that the sum of the two numbers is 16. Promotional Contest Use the table in Exercise 15 to less than 8? find the probability that the winner entered on Monday or 8. What is the probability that the sum of the two numbers is Saturday. greater than 4? 17. Find P͑A ʜ B͒ if P͑A͒ ϭ 0.2, P͑B͒ ϭ 0.5 and 9. What is the probability that the product of the two numbers P͑A ʝ B͒ ϭ 0.1. is 2 or 3? 18. Find P͑A ʝ B͒ if P͑A͒ ϭ 1͞2, P͑B͒ ϭ 2͞3 and 10. What is the probability that the product of the two numbers P͑A ʜ B͒ ϭ 5͞6. is 2, 3, or 7? 19. Find P͑A͒ if P͑B͒ ϭ 1͞3, P͑A ʜ B͒ ϭ 2͞3 and P͑A ʝ B͒ ϭ 1͞9. 11. State Lottery In a state lottery, a random number is selected between 000 and 999. What is the probability that 20. Find P͑B͒ if P͑A͒ ϭ 0.5, P͑A ʜ B͒ ϭ 0.8 and the number is less than 100 or greater than 800? P͑A ʝ B͒ ϭ 0. 12. Birthday If a birthdate is selected at random from Card Game In Exercises 21–24, one card is selected among 365 days, what is the probability that it is a date in from a standard deck of 52 playing cards. Find the May or June? indicated probability. 13. Blood Type In a sample of 24 hospital patients, 9 of 21. The probability that the card is either a face card or a club. them have type O blood, 6 have type A, 5 have type B, and 4 have type AB. If a patient is chosen at random from the 22. The probability that the card is an ace or a diamond. sample, what is the probability that the patient’s blood is 23. The probability that the card is black or a face card. type A or type B? 24. The probability that the card is red or a king. APPENDIX C.2 Probability Computations A17

Dice In Exercises 25–28, two six-sided dice are tossed. 36. Random Selection A dormitory has 60 students, 24 of Find the indicated probability. whom are taking an English test, and 16 of whom are tak- 25. The probability that the sum is odd or more than 8. ing a math test. Of the students taking tests, 8 are taking both tests. What is the probability that a student chosen at 26. The probability that the sum is even or prime. random from the dormitory is taking a test? 27. The probability that the product is 6 or the sum is 7. 37. Defective Units The probability of getting at least one 28. The probability that the product is 3 or the sum is even. defective unit in a shipment of ten units is p. What is the probability that all ten are nondefective? 29. Telephone Number If a computer generates the first 38. Defective Units The probability of getting at least two digit of a telephone number at random, what is the proba- defective units in a shipment of ten units is 0.30. What is bility that the digit is even or less than 4? the probability that at least nine of the units are nondefec- 30. Election Poll Three people have been nominated for tive? president of a college class. From a small poll, it is estimat- 39. Probability of a Complement The management of a ed that the probability of Candidate I winning the election company plans to select one of its employees at random to is 0.37, and the probability of Candidate II winning the feature in the company’s next quarterly report. The proba- election is 0.44. What is the probability that the third can- bility of selecting a female employee is 2͞9. What is the didate will win the election? probability of selecting a male employee? 31. Random Selection A boat builder assembles sailboats 40. Probability of a Complement An auditorium is filled and motorboats at four locations (A, B, C, D) as shown in with 500 people, 150 of whom are 40 years old or older. the table. What is the probability that a boat selected at What is the probability that someone selected at random random is a motorboat or was assembled at location D? from the auditorium is under 40 years old? 41. Probability of a Complement If the only way to win A B C D a card game is to select a face card or an ace from a stan- Sailboats 16 9 4 10 dard deck of cards, what is the probability of losing the game? Motorboats 10 7 9 5 42. Probability of a Complement There are 48 cell- phones at a phone store, 20 of which are flip phones. What 32. Random Selection Use the table in Exercise 31 to find is the probability that a cell phone chosen at random is not the probability that a boat selected at random is a sailboat a flip phone? or was assembled at location B. In Exercises 43–46, find the odds of an event with the Random Selection In Exercises 33 and 34, a single given probability. ball is drawn at random from a container filled with 43.P͑A͒ ϭ 1 44. P͑A͒ ϭ 1 50 balls, 25 red numbered from 1–25 and 25 black 6 2 ͑ ͒ ϭ 2 ͑ ͒ ϭ 3 numbered from 1–25. 45.P A 5 46. P A 10 33. What is the probability that a red or even-numbered ball is 47. What are the odds of guessing the month of a stranger’s drawn? birthdate? 34. What is the probability that a black ball or a ball numbered 48. What are the odds of drawing a club or a red card from a less than 10 is drawn? standard deck of 52 playing cards? 35. Random Selection A section of a forest is composed In Exercises 49 and 50, the odds that an event will of several species of trees as shown in the table. What is the occur are given. Find the probability (a) that the event probability that a tree cut down at random is a height that will occur, and (b) that the event will not occur. is less than 50 feet or is an oak? 49. 4 to 1 50. 3 to 2 Height Pine Oak Birch 51. Probability of Winning A statistician figures that the Less than 50 feet 17 21 10 odds of a horse winning a race are 2 to 5. Based on these odds, what is the probability that the horse will win the 50 feet or more 20 16 16 race? 52. Probability of Winning The odds of the home team winning a football game are 1 to 3. What is the probability that the visiting team will win?