Distributions. and Probability. Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment


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1 C Probability and Probability Distributions APPENDIX C.1 Probability A1 C.1 Probability Find the sample space of an experiment. Find the probability of an event. Sample Space of an Experiment When assigning measurements to the uncertainties of everyday life, people often use ambiguous terminology such as fairly certain, probable, and highly unlikely. Probability theory allows you to remove this ambiguity by assigning a number to the likelihood of the occurrence of an event. This number is called the probability that the event will occur. For example, if you toss a fair coin, the probability that it will land heads up is onehalf or 0.5. In probability theory, any happening whose result is uncertain is called an experiment. Each repetition of an experiment is called a trial. The possible results of the experiment are outcomes, the set of all possible outcomes of the experiment is the sample space of the experiment, and any subcollection of a sample space is an event. For instance, when a sixsided die is tossed, the sample space can be represented by the numbers 1 through 6. For this experiment, each of the outcomes is equally likely. To describe sample spaces in such a way that each outcome is equally likely, you must sometimes distinguish between various outcomes in ways that appear artificial. Example 1 illustrates such a situation. Example 1 Finding Sample Spaces Find the sample space for each of the following. a. One coin is tossed. b. Two coins are tossed. SOLUTION CHECKPOINT 1 An experiment consists of tossing a coin and a sixsided die. Find the sample space for the experiment. a. Because the coin will land heads up (denoted by H) or tails up (denoted by T), the sample space is S H, T. b. Because either coin can land heads up or tails up, the possible outcomes are as follows. HH heads up on both coins HT heads up on first coin and tails up on second coin TH tails up on first coin and heads up on second coin TT tails up on both coins So, the sample space is S HH, HT, TH, TT. Note that the list distinguishes between the two cases HT and TH, even though these two outcomes appear to be similar.
2 A2 APPENDIX C Probability and Probability Distributions The Probability of an Event To calculate the probability of an event, count the number of outcomes in the event and in the sample space. The number of equally likely outcomes in event E is denoted by n E, and the number of equally likely outcomes in the sample space S is denoted by n S. The probability that event E will occur is given by n E n S. Increasing likelihood of occurrence The Probability of an Event If an event E has n E equally likely outcomes and its sample space S has n S equally likely outcomes, then the probability of event E is n S. Impossible event (cannot occur) FIGURE C.1 The occurrence of the event is just as likely as it is unlikely. Certain event (must occur) Because the number of outcomes in an event must be less than or equal to the number of outcomes in the sample space, the probability of an event must be a number between 0 and 1. That is, 0 P E 1 as indicated in Figure C.1. If P E 0, event E cannot occur, and E is called an impossible event. If P E 1, event E must occur, and E is called a certain event. Example 2 Finding Probabilities of Events Find the probability of the following events. a. Two coins are tossed. What is the probability that both land heads up? b. A card is drawn from a standard deck of playing cards. What is the probability that it is an ace? SOLUTION STUDY TIP You can write a probability as a fraction, decimal, or percent. For instance, in Example 2(a), the probability of getting two 1 heads can be written as 4, 0.25, or 25%. a. Following the procedure in Example 1(b), let E HH and S HH, HT, TH, TT. The probability of getting two heads is n S 1 4. b. Because there are 52 cards in a standard deck of playing cards and there are 4 aces (one in each suit), the probability of drawing an ace is n S CHECKPOINT 2 A card is drawn from a standard deck of playing cards. What is the probability that it is a face card (king, queen, or jack)?
3 APPENDIX C.1 Probability A3 Example 3 Finding the Probability of an Event Two sixsided dice are tossed. What is the probability that the sum of the dice is 7? (See Figure C.2.) SOLUTION Because there are 6 possible outcomes on each die, you can use the Fundamental Counting Principle to conclude that there are 6 6 or 36 different outcomes when two dice are tossed. To find the probability of rolling a sum of 7, you must first count the number of ways in which this can occur. FIGURE C.2 First die Second die Algebra Review For examples on how to count the number of ways an event can happen, see the Chapter 11 Algebra Review on pages 691 and 692. So, a total of 7 can be rolled in 6 ways, which means that the probability of rolling a 7 is n S CHECKPOINT 3 One sixsided die is tossed twice. What is the probability that the sum of the two tosses is 4? Example 4 Finding the Probability of an Event FIGURE C.3 Twelvesided dice, as shown in Figure C.3, can be constructed (in the shape of regular dodecahedrons) such that each of the numbers from 1 to 6 appears twice on each die. Show that these dice can be used in any game requiring ordinary sixsided dice without changing the probabilities of different outcomes. SOLUTION For an ordinary sixsided die, each of the numbers 1, 2, 3, 4, 5, and 6 occurs only once, so the probability of any particular number is n S 1 6. For one of the twelvesided dice, each number occurs twice, so the probability of any particular number is n S CHECKPOINT 4 One twelvesided die is tossed twice. What is the probability that the sum of the two tosses is 11?
4 A4 APPENDIX C Probability and Probability Distributions Example 5 The Probability of Winning a Lottery In a state lottery, a player chooses 6 different numbers from 1 to 40. If these six numbers match the six numbers drawn (in any order) by the lottery commission, the player wins (or shares) the top prize. What is the probability of winning the top prize if the player buys one ticket? SOLUTION Because the order of the numbers is not important, use the formula for the number of combinations of 40 elements taken 6 at a time to determine the size of the sample space. n S 40 C 6 If a person buys only one ticket, the probability of winning the top prize is n S 1 3,838, ,838,380 CHECKPOINT 5 A bag contains two green, three yellow, and four red marbles. If two marbles are drawn from the bag without replacement, what is the probability that both marbles are red? CONCEPT CHECK 1. What is an experiment? 2. What is a sample space? 3. To determine the of an event, you can use the formula n S, where n E is the number of equally likely outcomes in the event and n S is the number of equally likely outcomes in the sample space. 4. If P E 0, then E is a(n) event, and if P E 1, then E is a(n) event.
5 APPENDIX C.1 Probability A5 Skills Review C.1 The following warmup exercises involve skills that were covered in earlier sections. You will use these skills in the exercise set for this section. For additional help, review the Chapter 11 Algebra Review on pages 691 and 692. In In Exercises 1 4, write the fraction (a) in lowest terms, (b) as a decimal, and (c) as a percent In Exercises 5 10, evaluate the expression. x 10x ! 8 C C ! 10 C ! 16 C 2 4 C 3 20 C 5 Exercises C.1 In Exercises 1 4, determine the sample space for the experiment. 1. A sixsided die is tossed twice and the sum is recorded. 2. A taste tester has to rank three varieties of yogurt, A, B, and C, according to preference. 3. Two marbles are selected from a bag containing two red marbles, two blue marbles, and one yellow marble. The color of each marble is recorded. 4. Two county supervisors are selected from five supervisors, A, B, C, D and E, to study a recycling plan. Tossing a Coin In Exercises 5 8, find the probability for the experiment of tossing a coin three times. Use the sample space S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. 5. The probability of getting exactly one tail 6. The probability of getting a head on the first toss 7. The probability of getting at least one head 8. The probability of getting at least two heads Drawing a Card In Exercises 9 12, find the probability for the experiment of selecting 1 card from a standard deck of 52 playing cards. 9. The card is black. 10. The card is a red face card. 11. The card is a 6 or lower. (Aces are low.) 12. The card is not a face card. See for workedout solutions to oddnumbered exercises. Tossing a Die In Exercises 13 18, find the probability for the experiment of tossing a sixsided die twice. 13. The sum is The sum is at least The sum is less than The sum is 2, 3, or The sum is no more than The sum is odd or prime. In Exercises 19 22, a computer generates an integer from 1 through 20 at random. Find the probability of the event. 19. The probability of generating a multiple of The probability of generating a number divisible by The probability of generating a prime number 22. The probability of generating a factor of 24 Drawing Marbles In Exercises 23 26, find the probability for the experiment of drawing two marbles (without replacement) from a bag containing one green, two yellow, and three red marbles. 23. Both marbles are red. 24. Both marbles are yellow. 25. Neither marble is yellow. 26. Neither marble is red.
6 A6 APPENDIX C Probability and Probability Distributions 27. Jury Selection A person is selected at random for jury duty from a list of registered voters from three different counties. Country A has 14,789 registered voters, County B has 17,851 registered voters, and County C has 23,487 registered voters. If only one name is selected, what is the probability that the person chosen is from County C? 28. Order of Arrival Three fire engines, four police cars, and one ambulance are called to the scene of an accident. If they all have an equal chance of arriving at the same time, what is the probability that a police car will arrive first? 29. Random Selection Nine players went to bat in the sixth inning of a baseball game. Four had singles, one had a double, one had a grand slam, and the others struck out. What is the probability that a batter chosen at random struck out? 30. Contract Bidding Ten health insurance companies are bidding for an insurance contract. Three are local companies, three have statewide operations, and four companies have national operations. Only one company will be awarded the contract. If each company is equally likely to win the contract, what is the probability that the contract will be awarded to one of the companies with statewide operations? 31. Data Analysis A study of the effectiveness of a flu vaccine was conducted with a sample of 500 people. Some participants in the study were given no vaccine, some were given one injection, and some were given two injections. The results of the study are listed in the table. No vaccine One injection Two injections Total Flu No flu Total A person is selected at random from the sample. Find the indicated probability (a) The person had two injections. (b) The person did not get the flu. (c) The person got the flu and had one injection. 32. Data Analysis One hundred college students were interviewed to determine their political party affiliations and whether they favored a balancedbudget amendment to the Constitution. The results of the study are listed in the table, where D represents Democrat and R represents Republican. Favor Not Favor Unsure Total D R Total A person is selected at random from the sample. Find the probability that the person described is selected. (a) A person who does not favor the amendment (b) A Republican (c) A Democrat who favors the amendment 33. Alumni Association A college sends a survey to selected members of the class of Of the 1254 people who graduated that year, 672 are women, of whom 124 went on to graduate school. Of the 582 male graduates, 198 went on to graduate school. An alumni member is selected at random. What are the probabilities that the person is (a) female, (b) male, and (c) female and did not attend graduate school? 34. Education In a high school graduating class of 202 students, 95 are on the honor roll. Of these, 71 are going on to college, and of the other 107 students, 53 are going on to college. A student is selected at random from the class. What are the probabilities that the person chosen is (a) on the honor roll, (b) going to college, and (c) on the honor roll, but not going to college? 35. Defective Item A clerk sold an equal number of hats, scarves, gloves, ski masks, and ear muffs. If one of the items was returned because it was defective, what is the probability that it was a hat? 36. Birth Order Each of six motherstobe received 3D ultrasound scans, which showed that four of them will give birth to girls. What is the probability that the first two women to give birth will have boys? 37. Random Selection Four letters and envelopes are addressed to four different people. If the letters are inserted into the envelopes at random, what is the probability that exactly one letter will be inserted in the correct envelope? 38. Random Selection A math teacher chooses 5 students at random from a class of 20 to solve a problem at the board. If 12 students know how to solve the problem, what is the probability that (a) all 5 students picked do not know how to solve the problem, and (b) exactly 3 students picked know how to solve the problem? 39. Defective Units A shipment of 12 microwave ovens contains 3 defective units. A vending company has ordered 4 of these units. Because the microwave ovens are identically packaged, the selection will be at random. What is the probability that (a) all 4 units are good, and (b) exactly 2 units are good?
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