The Probability of an Event

Transcription

1 PPENDIX D Counting Principles and Probability D11 D. Probability The Probability of an Event Using Counting Methods to Find Probabilities The Probability of an Event The probability of an event is a number from 0 to 1 that indicates the likelihood that the event will occur. n event that is certain to occur has a probability of 1. n event that cannot occur has a probability of 0. n event that is equally likely 1 to occur or not occur has a probability of or 0.5. Probability of 0: Event cannot occur. Probability of 0.5: Event is equally likely to occur or not occur. Probability of 1: Event must occur The Probability of an Event Consider a sample space S that is composed of a finite number of outcomes, each of which is equally likely to occur. subset E of the sample space is an event. The probability that an outcome E will occur is number of outcomes in event number of outcomes in sample space. EXMPLE 1 Finding the Probability of an Event (a) You are dialing a friend s phone number but cannot remember the last digit. If you choose a digit at random, the probability that it is correct is (b) On a multiple-choice test, you know that the answer to a question is not (a) or (d), but you are not sure about (b), (c), and (e). If you guess, the probability that you are wrong is number of correct digits number of possible digits number of wrong answers number of possible answers.

2 D1 PPENDIX D Counting Principles and Probability EXMPLE Conducting a Poll The Centers for Disease Control surveyed 11,61 high school students. The students were asked whether they considered themselves to be a good weight, underweight, or overweight. The results are shown in Figure D.1. Female Good weight 08 Overweight 1776 Good weight 89 Male Underweight 66 Overweight 961 Underweight 1057 FIGURE D.1 (a) If you choose a female at random from those surveyed, the probability that she said she was underweight is number of females who answered "underweight" number of females in survey (b) If you choose a person who answered underweight from those surveyed, the probability that the person is female is number of females who answered "underweight" number in survey who answered "underweight" well-known experiment in probability is called the birthday problem. This problem concerns the probability that in a group of n people at least two of the people will have the same birthday. program that simulates the birthday problem for several models of calculators can be found in ppendix H.

3 PPENDIX D Counting Principles and Probability D1 Polls such as the one described in Example are often used to make inferences about a population that is larger than the sample. For instance, from Example, you might infer that 7% of all high school girls consider themselves to be underweight. When you make such an inference, it is important that those surveyed are representative of the entire population. Drain } in. 6 in. EXMPLE Using rea to Find Probability You have just stepped into the tub to take a shower when one of your contact lenses falls out. (You have not yet turned on the water.) ssuming the lens is equally likely to land anywhere on the bottom of the tub, what is the probability that it lands in the drain? Use the dimensions in Figure D.. FIGURE D. 50 in. Because the area of the tub bottom is square inches and the area of the drain is 1 rea of drain r square inches, the probability that the lens lands in the drain is BC Bc bc bc EXMPLE The Probability of Inheriting Certain Genes Common parakeets have genes that can produce any one of the following four feather colors. BC Bc bc BBCC BBCc BbCC BBCc BBcc BbCc BbCC BbCc bbcc BbCc Bbcc bbcc Green: BBCC, BBCc, BbCC, BbCc Blue: BBcc, Bbcc Yellow: bbcc, bbcc White: bbcc Use the Punnett square in Figure D. to find the probability that an offspring to two green parents (both with BbCc feather genes) will be yellow. Note that each parent passes along a B or b gene and a C or c gene. bc BbCc Bbcc bbcc bbcc The probability that an offspring will be yellow is FIGURE D. number of yellow possibilities number of possibilities 16. If P is the probability that an outcome will occur, the probability that the outcome will not occur is 1 P. In Example, the probability that the lens does not land in the drain is approximately , or In Example, the probability that an offspring will not be yellow is 1 16, or 1 16.

4 D1 PPENDIX D Counting Principles and Probability Standard 5-Card Deck K Q J K Q J FIGURE D. K Q J K Q J Using Counting Methods to Find Probabilities In the following examples, you will see how the basic counting principles you learned in the previous section are used to determine the probability that an event will occur. EXMPLE 5 The Probability of a Royal Flush Five cards are dealt at random from a standard deck of 5 playing cards (see Figure D.). What is the probability that the cards are 10-J-Q-K- of the same suit? From Example 11 on page D7, you saw that the number of possible five-card hands from a deck of 5 cards is 5 C 5,598,960. Because only four of these five-card hands are 10-J-Q-K- of the same suit, the probability that the hand contains these cards is,598, ,70. Getting Costumed Created 60 FIGURE D.5 Borrowed 0 Rented 60 Bought 60 EXMPLE 6 Conducting a Survey survey was conducted of 500 adults who had worn Halloween costumes. Each person was asked how he or she acquired a Halloween costume: created it, rented it, bought it, or borrowed it. The results are shown in Figure D.5. What is the probability that the first four people who were polled all created their costumes? To answer this question, you need to use the formula for the number of combinations twice. First, find the number of ways to choose four people from the 60 who created their own costumes. 60 C Next, find the number of ways to choose four people from the 500 who were surveyed. 500 C The probability that all of the first four people surveyed created their own costumes is the ratio of these two numbers number of ways to choose from 60 number of ways to choose from ,5,10,57,01, ,5,10,57,01,15

5 PPENDIX D Counting Principles and Probability D15 EXMPLE 7 Forming a Committee To obtain input from 00 company employees, the management of a company selected a committee of five. Of the 00 employees, 56 were from minority groups. None of these 56, however, was selected to be on the committee. Does this indicate that the management s selection was biased? Part of the solution is similar to that of Example 6. If the five committee members were selected at random, the probability that all five would be nonminority is 1C 5 00 C 5 81,008,58,55,650, number of ways to choose 5 from 1 nonminority employees number of ways to choose 5 from 00 employees So, if the committee were chosen at random (that is, without bias), the likelihood that it would have no minority members is about lthough this does not prove that there was bias, it does suggest it. D. Exercises Sample Space In Exercises 1, determine the number of outcomes in the sample space for the experiment. 1. One letter from the alphabet is chosen.. six-sided die is tossed twice and the sum is recorded.. Two county supervisors are selected from five supervisors,, B, C, D, and E, to study a recycling plan.. salesperson makes a presentation about a product in three homes. In each home, there may be a sale (denote by Y) or there may be no sale (denote by N). Sample Space In Exercises 5 8, list the outcomes in the sample space for the experiment. 5. taste tester must taste and rank three brands of yogurt,, B, and C, according to preference. 6. coin and a die are tossed. 7. basketball tournament between two teams consists of three games. For each game, your team may win (denote by W) or lose (denote by L). 8. Two students are randomly selected from four students,, B, C, and D. In Exercises 9 and 10, you are given the probability that an event will occur. Find the probability that the event will not occur. 9. p p 0.8 In Exercises 11 and 1, you are given the probability that an event will not occur. Find the probability that the event will occur. 11. p p 0.1

6 D16 PPENDIX D Counting Principles and Probability Coin Tossing In Exercises 1 16, a coin is tossed three times. Find the probability of the specified event. Use the sample space S HHH, HHT, HTH, HTT, 1. The event of getting two heads 1. The event of getting a tail on the second toss 15. The event of getting at least one head 16. The event of getting no more than two heads Playing Cards In Exercises 17 0, a card is drawn from a standard deck of 5 playing cards. Find the probability of drawing the indicated card. 17. red card 18. queen 19. face card 0. black face card Tossing Die In Exercises 1, a six-sided die is tossed. Find the probability of the specified event. 1. The number is a 5.. The number is a 7.. The number is no more than 5.. The number is at least 1. United States Blood Types In Exercises 5 and 6, use the circle graph, which shows the percent of people in the United States in 001 with each blood type. (Source: merica s Blood Centers) Type 0% Type B 11% THH, THT, TTH, TTT. United States Blood Types Type B % Type O 5% 5. person is selected at random from the United States population. What is the probability that the person does not have blood type B? 6. What is the probability that a person selected at random from the United States population does have blood type B? How is this probability related to the probability found in Exercise 5? College Freshmen In Exercises 7 0, use the circle graphs, which show the percent of incoming college freshmen in the United States for each average high school grade category for the years 1990 and (Source: The Higher Education Research Institute) % C 19% 1990 B 58% B 5% 1999 % C 1% 7. person is selected at random from the 1990 freshman class. What is the probability that the person s average high school grade was an? 8. person is selected at random from the 1999 freshman class. What is the probability that the person s average high school grade was an? 9. person is selected at random from the 1990 freshman class. What is the probability that the person s average high school grade was not a C? 0. person is selected at random from the 1999 freshman class. What is the probability that the person s average high school grade was not a B? 1. Multiple-Choice Test student takes a multiplechoice test in which there are five choices for each question. Find the probability that the first question is answered correctly given the following conditions. (a) The student has no idea of the answer and guesses at random. (b) The student can eliminate two of the choices and guesses from the remaining choices. (c) The student knows the answer.. Multiple-Choice Test student takes a multiplechoice test in which there are four choices for each question. Find the probability that the first question is answered correctly given the following conditions. (a) The student has no idea of the answer and guesses at random. (b) The student can eliminate two of the choices and guesses from the remaining choices. (c) The student knows the answer.

7 PPENDIX D Counting Principles and Probability D17. Random Selection Twenty marbles numbered 1 through 0 are placed in a bag, and one is selected. Find the probability of the specified event. (a) The number is 1. (b) The number is prime. (c) The number is odd. (d) The number is less than 6.. Class Election Three people are running for class president. It is estimated that the probability that Candidate will win is 0.5 and the probability that Candidate B will win is 0.. (a) What is the probability that either Candidate or Candidate B will win? (b) What is the probability that the third candidate will win? In Exercises 5 6, some of the sample spaces are large; therefore, you should use the counting principles discussed in the previous section. 5. Game Show On a game show, you are given five digits to arrange in the proper order for the price of a car. If you arrange them correctly, you win the car. Find the probability of winning if you know the correct position of only one digit and must guess the positions of the other digits. 6. Game Show On a game show, you are given four digits to arrange in the proper order for the price of a grandfather clock. What is the probability of winning given the following conditions? (a) You guess the position of each digit. (b) You know the first digit, but must guess the remaining three. 7. Lottery You buy a lottery ticket inscribed with a five-digit number. On the designated day, five digits (0 though 9 inclusive) are randomly selected. What is the probability that you have a winning ticket? 8. Shelving Books parent instructs a child to place a five-volume set of books on a bookshelf. Find the probability that the books are placed in the correct order if the child places them at random. 9. Preparing for a Test n instructor gives her class a list of 10 study problems, from which she will select 8 to be answered on an exam. You know how to solve eight of the problems. What is the probability that you will be able to answer all eight questions on the exam? 0. Committee Selection committee of three students is to be selected from a group of three girls and five boys. Find the probability that the committee is composed entirely of girls. 1. Defective Units shipment of 10 food processors to a certain store contains defective units. You purchase two of these food processors as birthday gifts for friends. Determine the probability that you get both defective units.. Defective Units shipment of 1 compact disc players contains defective units. husband and wife buy three of these compact disc players to give to their children as birthday gifts. (a) What is the probability that none of the three units is defective? (b) What is the probability that at least one unit is defective?. Book Selection Four books are selected at random from a shelf containing six novels and four autobiographies. Find the probability that the four autobiographies are selected.. Card Selection Five cards are selected from a standard deck of 5 cards. Find the probability that four aces are selected. 5. Card Selection Five cards are selected from a standard deck of 5 cards. Find the probability that all hearts are selected. 6. Card Selection Five cards are selected from a standard deck of 5 cards. Find the probability that two aces and three queens are selected. 7. Girl or Boy? The genes that determine the sex of humans are denoted by XX (female) and XY (male). Male X Y Female X X???? (a) Complete the Punnett square in the figure. (b) What is the probability that a newborn will be a girl? a boy? (c) Explain why it is equally likely that a newborn baby will be a boy or a girl.

8 D18 PPENDIX D Counting Principles and Probability 8. Blood Types There are four basic human blood types: ( or o), B (BB or Bo), B (B), and O (oo). B o? o??? (a) Complete the Punnett square in the figure. (b) What is the blood type of each parent? (c) What is the probability that their offspring will have blood type? B? B? O? 51. Study Questions n instructor gives the class a list of four study questions for the next exam. Two of the four study questions will be on the exam. Find the probability that a student who knows the material relating to three of the four questions will be able to answer both questions selected for the exam. 5. Geometry child uses a spring-loaded device to shoot a marble into the square box shown in the figure. The base of the square is horizontal and the marble is equally likely to come to rest at any point on the base. In parts (a) (d), find the probability that the marble comes to rest in the specified region. 9. Meeting Time You and a friend agree to meet at a favorite restaurant between 5:00 P.M. and 6:00 P.M. The one who arrives first will wait 15 minutes for the other, after which the first person will leave (see figure). What is the probability that the two of you actually meet, assuming that your arrival times are random within the hour? Your friend s arrival time (in minutes past 5:00..) PM 60 You meet You meet You don't meet You arrive first Your friend arrives first Your arrival time (in minutes past 5:00 PM..) 50. Continuing Education In a high school graduating class of 5 students, 55 are going to continue their education. What is the probability that a student selected at random from the class will not be continuing his or her education? in. in. (a) The red center (b) The blue ring (c) The purple border (d) Not in the yellow ring 5. Think bout It The probability of an event is. What is the probability that the event does not occur? Explain. 5. Think bout It The weather forecast indicates that the probability of rain is 0%. Explain what this means.