PROBABILITY section. The Probability of an Event


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1 4.3 Probability (43) PROBABILITY In this section In the two preceding sections we were concerned with counting the number of different outcomes to an experiment. We now use those counting techniques to find probabilities. The Probability of an Event The Addition Rule Complementary Events Odds The Probability of an Event In probability an experiment is a process such as tossing a coin, tossing a die, drawing a poker hand from a deck, or arranging people in a line. A sample space is the set of all possible outcomes to an experiment. An event is a subset of a sample space. For example, if we toss a coin, then the sample space consists of two equally likely outcomes, heads and tails. We write S H, T. The subset E H is the event of getting heads when the coin is tossed. We use n(s) to represent the number of equally likely outcomes in the sample space S and n(e ) to represent the number of outcomes in the event E. For the example of tossing a coin, n(s) 2 and n(e). The Probability of an Event If S is a sample space of equally likely outcomes to an experiment and the event E is a subset of S, then the probability of E, P(E), is defined to be n(e) P(E). n(s) When S H, T and E H, n(e) P(E) 2. n(s) So the probability of getting heads on a single toss of a coin is. 2 If E is the event of getting 2 heads on a single toss of a coin, then n(e) 0 and P(E) If E is the event of getting fewer than 2 heads on a single toss of a coin, then for either outcome H or T we have fewer than 2 heads. So E H, T, n(e) 2, and P(E) 2 2. Note that the probability of an event is a number between 0 and inclusive, being the probability of an event that is certain to occur and 0 being the probability of an event that is impossible to occur. E X A M P L E Rolling a die What is the probability of getting a number larger than 4 when a single die is rolled? When we roll a die, we count the number of dots showing on the upper face of the die. So the sample space of equally likely outcomes is S, 2, 3, 4, 5, 6. Since only 5 and 6 are larger than 4, E 5, 6. According to the definition of probability, n(e) P(E) n(s)
2 728 (44) Chapter 4 Counting and Probability M A T H A T W O R K The probability experiments discussed in this chapter are not just textbook examples that have no relationship to real life. For example, if a couple plans to have 6 children and the probability of having a girl on each try is 2, then the couple can expect to have 3 girls. If you guess at the answer to each question of a 00question, 5choice multiplechoice test, then you have 5 probability of getting each question correct, and you can expect to get 20 questions correct. Try it. The expected number of successes is the product of the probability of success and the number of tries. Lotteries provide us an opportunity to observe massive probability experiments. In the Florida Lottery you can win by choosing 6 numbers from the numbers through 49 and matching the 6 numbers chosen by the Florida Lottery. There are C(49, 6) ways to choose 6 numbers from 49, so the probability of winning on any individual try is. C(49, 6) 3,98 3,86 In the fall of 990 the weekly drawing frequently had relatively few participants, and consequently there was no winner for many weeks. When the prize got up to $06.5 million, the lottery got national attention. People came from everywhere to participate. During the week prior to September 5, 990, 09,63,978 tickets were sold. We expected 3,98 09,63, winners. On September 5 the 3,86 winning numbers were announced, and 6 winners shared the prize. Of course, probability cannot predict the future like a fortuneteller, but the power of probability to tell us what to expect is truly amazing. LOTTERIES E X A M P L E 2 Tossing coins What is the probability of getting at least one head when a pair of coins is tossed? Since there are 2 equally likely outcomes for the first coin and 2 equally likely outcomes for the second coin, by the fundamental counting principle there are 4 equally likely outcomes to the experiment of tossing a pair of coins. We can list the outcomes as ordered pairs: S (H, H), (H, T), (T, H), (T, T). Since 3 of these outcomes result in at least one head, E (H, H), (H, T), (T, H), and n(e) 3. So n(e) P(E) 3 4. n(s) E X A M P L E 3 Rolling a pair of dice What is the probability of getting a sum of 6 when a pair of dice is rolled?
3 Since there are 6 equally likely outcomes for each die, there are equally likely outcomes to the experiment of rolling the pair. We can list the outcomes as ordered pairs: S (, ), (, 2), (, 3), (, 4), (, 5), (, 6), (2, ), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, ), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, ), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, ), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, ), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) 4.3 Probability (45) 729 The sum of the numbers is 6, describes the event E (5, ), (4, 2), (3, 3), (2, 4), (, 5). So n(e) 5 P(E). n(s) The Addition Rule In tossing a pair of dice, let A be the event that doubles occurs and B be the event that the sum is 4. We can list the following events and their probabilities: 6 A (, ), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) and P(A) 3 B (3, ), (2, 2), (, 3) and P(B) A B (, ), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (3, ), (, 3) and P(A B) 3 86 A B (2, 2) and P(A B) 8 Note that the probability of doubles or a sum of 4, P(A B), is and This equation makes sense because there is one outcome, (2, 2), that is in both the events A and B. This example illustrates the addition rule. The Addition Rule If A and B are any events in a sample space, then P(A B) P(A) P(B) P(A B). If P(A B) 0, then A and B are called mutually exclusive events and P(A B) P(A) P(B). Note that for mutually exclusive events it is impossible for both events to occur. The addition rule for mutually exclusive events is a special case of the general addition rule.
4 730 (46) Chapter 4 Counting and Probability E X A M P L E 4 E X A M P L E 5 The addition rule At Downtown College 60% of the students are commuters (C), 50% are female (F), and 30% are female commuters. If a student is selected at random, what is the probability that the student is either a female or a commuter? By the addition rule the probability of selecting either a female or a commuter is P(F C) P(F) P(C) P(F C) The addition rule with dice In rolling a pair of dice, what is the probability that the sum is 2 or at least one die shows a 2? Let A be the event that the sum is 2 and B be the event that at least one die shows a 2. Since A occurs on only one of the equally likely outcomes (see Example 3), P(A). Since B occurs on of the equally likely outcomes, P(B). Since A and B are mutually exclusive, we have P(A B) P(A) P(B) 2 3. Complementary Events If the probability of rain today is 60%, then the probability that it does not rain is 40%. Rain and not rain are called complementary events. There is no possibility that both occur, and one of them must occur. If A is an event, then A (read A bar or A complement ) represents the complement of the event A. Note that complementary events are mutually exclusive, but mutually exclusive events are not necessarily complementary. Complementary Events Two events A and A are called complementary events if A A and P(A) P(A). E X A M P L E 6 E X A M P L E 7 Complementary events What is the probability of getting a number less than or equal to 4 when rolling a single die? We saw in Example that getting a number larger than 4 when rolling a single die has probability 3. The complement to getting a number larger than 4 is getting a number less than or equal to 4. So the probability of getting a number less than or equal to 4 is 2 3. Complementary events If the probability that White Lightning will win the Kentucky Derby is 0.5, then what is the probability that White Lightning does not win the Kentucky Derby?
5 4.3 Probability (47) 73 study tip Study for the final exam by working actual test questions. Be sure to rework all of your tests. Do the chapter tests in this book. You can get more tests to work by asking students or instructors for tests that were given in other classes of this course. Let W be winning the Kentucky Derby and N be not winning the Kentucky Derby. Since W and N are complementary events, we have P(W) P(N). So P(N) P(W) Odds If the probability is 2 3 that the Giants win the Super Bowl and 3 that they lose, then they are twice as likely to win as they are to lose. We say that the odds in favor of the Giants winning the Super Bowl are 2 to. Notice that odds are not probabilities. Odds are ratios of probabilities. We usually write odds as ratios of whole numbers. Odds If A is any event, then the odds in favor of A is the ratio P(A) to P(A) and the odds against A is the ratio of P(A) to P(A). E X A M P L E 8 Determining odds What are the odds in favor of getting a sum of 6 when rolling a pair of dice? What are the odds against a sum of 6? 5 In Example 3 we found the probability of a sum of 6 to be. So the probability of the complement (the sum is not 6) is 3. The odds in favor of getting a sum of 6 are 5 to 3. Multiply each fraction by to get the odds 5 to 3. The odds against a sum of 6 are 3 to 5. E X A M P L E 9 helpful hint Odds and probability are often confused, even by people who write lottery tickets. If the probability of winning a lottery is, then the probability of losing is, and the 00 odds in favor of winning are to 99. Many lottery tickets will state (incorrectly) that the odds in favor of winning are to 00. Determining probability given the odds If the odds in favor of Daddy s Darling winning the third race at Delta Downs are 4 to, then what is the probability that Daddy s Darling wins the third race? Since 4 to is the ratio of the probability of winning to not winning, the probability of winning is four times as large as the probability of not winning. Let P(W) x and P(W ) 4x. Since P(W ) P(W ), we have 4x x, or 5x, or x 5. So the probability of winning is 4 5. We can write the idea found in Example 9 as a strategy for converting from odds to probabilities. Strategy for Converting from Odds to Probability If the odds in favor of event E are a to b, then a b P(E) and P(E). a b a b
6 732 (48) Chapter 4 Counting and Probability WARMUPS True or false? Explain your answer.. If S is a sample space of equally likely outcomes and E is a subset of S, then P(E) n(e). False 2. If an experiment consists of tossing 3 coins, then the sample space consists of 6 equally likely outcomes. False 3. The probability of getting at least one tail when a coin is tossed twice is True 4. The probability of getting at least one 4 when a pair of dice is tossed is. True 5. The probability of getting at least one head when 5 coins are tossed is False 6. If 3 coins are tossed, then getting exactly 3 heads and getting exactly 3 tails are complementary events. False 7. If the probability of getting exactly 3 tails in a toss of 3 coins is 8, then the probability of getting at least one head is 7 8. True 8. If the probability of snow today is 80%, then the odds in favor of snow are 8 to 0. False 9. If the odds in favor of an event E are 2 to 3, then P(E) 2 3. False 0. The ratio of 2 to 3 is equivalent to the ratio of 2 to 3. False 4. 3 EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is an experiment? An experiment is a process for which the outcomes are uncertain. 2. What is a sample space? A sample space is the set of all possible outcomes to an experiment. 3. What is an event? An event is a subset of a sample space. 4. What is the probability of an event? The probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space. 5. What is the addition rule? The addition rule states that if A and B are events in a sample space, then P(A B) P(A) P(B) P(A B). 6. What are the odds in favor of an event? The odds in favor of an event is the ratio of the probability of the event to the probability of the complement of the event. Solve each probability problem. See Example If a single die is tossed, then what is the probability of getting a) a number larger than 3? b) a number less than or equal to 5? c) a number other than 6? d) a number larger than 7? e) a number smaller than 9? 2, 5 6, 5 6,0, 8. If a single coin is tossed once, then what is the probability of getting a) tails? c) exactly three heads? b) fewer than two heads? 2,,0 9. If a pair of coins is tossed, then what is the probability of getting a) exactly two heads? c) exactly two tails? b) at least one tail? d) at most one tail? 4, 3 4, 4, 3 4
7 4.3 Probability (49) If a single coin is tossed twice, then what is the probability of getting a) heads followed by tails? c) a tail on the second toss? b) two heads in a row? d) exactly one tail? 4, 4, 2, 2. If a pair of dice is tossed, then what is the probability of getting a) a pair of 2 s? d) a sum greater than? b) at least one 2? e) a sum less than 2? c) a sum of 7?, 6,,,0 2. If a single die is tossed twice, then what is the probability of getting a) a followed by a 2? b) a sum of 3? 8 c) a 6 on the second toss? 6 d) no more than two 5 s? e) an even number followed by an odd number? 4 3. A ball is selected at random from a jar containing 3 red balls, 4 yellow balls, and 5 green balls. What is the probability that a) the ball is red? 4 b) the ball is not yellow? 2 3 c) the ball is either red or green? 2 3 d) the ball is neither red nor green? 3 FIGURE FOR EXERCISE 3 4. A committee consists of Democrat, 5 Republicans, and 6 independents. If one person is randomly selected from the committee to be the chairperson, then what is the probability that a) the person is a Democrat? 2 b) the person is either a Democrat or a Republican? 2 7 c) the person is not a Republican? 2 5. A jar contains 0 balls numbered through 0. Two balls are randomly selected one at a time without replacement. What is the probability that a) is selected first and 2 is selected second? 9 0 b) the sum of the numbers selected is 3? c) the sum of the numbers selected is 6? A small company consists of a president, a vicepresident, and 4 salespeople. If 2 of the 6 people are randomly selected to win a Hawaiian vacation, then what is the probability that none of the salespeople is a winner? If a 5card poker hand is drawn from a deck of 52, then what is the probability that a) the hand contains the ace, king, queen, jack, and ten of spades? 2,59 8,960 b) the hand contains one 2, one 3, one 4, one 5, and one 6? 024 2,598, If 5 people with different names and different weights randomly line up to buy concert tickets, then what is the probability that a) they line up in alphabetical order? 20 b) they line up in order of increasing weight? 20 Use the addition rule to solve each problem. See Examples 4 and Among the drivers insured by American Insurance, 65% are women, 38% of the drivers are in a highrisk category, and 24% of the drivers are highrisk women. If a driver is randomly selected from that company, what is the probability that the driver is either highrisk or a woman? What is the probability of getting either a sum of 7 or at 5 least one 4 in the toss of a pair of dice? 2 2. A couple plans to have 3 children. Assuming males and females are equally likely, what is the probability that they have either 3 boys or 3 girls What is the probability of getting a sum of 0 or a sum of 5 7 in the toss of a pair of dice? 23. What is the probability of getting either a heart or an ace 4 when drawing a single card from a deck of 52 cards? What is the probability of getting either a heart or a spade when drawing a single card from a deck of 52 cards? 2 Solve each problem. See Examples 6 and If the probability of surviving a headon car accident at 55 mph is 0.005, then what is the probability of not surviving? If the probability of a tax return not being audited by the IRS is 0.97, then what is the probability of a tax return being audited? A pair of dice is tossed. What is the probability of a) getting a pair of 4 s? b) not getting a pair of 4 s? 3 5 c) getting at least one number that is not a 4? Three coins are tossed. What is the probability of a) getting three heads? 8 b) not getting three heads? 7 8 c) getting at least one tail? 7 8
8 734 (420) Chapter 4 Counting and Probability Solve each problem. See Examples 8 and If the probability is 60% that the eye of Hurricane Edna comes ashore within 30 miles of Charleston, then what are the odds in favor of the eye of Edna coming ashore within 30 miles of Charleston? 3 to 2 FIGURE FOR EXERCISE 38 FIGURE FOR EXERCISE If the probability that a Sidewinder missile hits its target is 8 9, then what are the odds a) in favor of the Sidewinder hitting its target? 8 to b) against the Sidewinder hitting its target? to 8 3. If the probability that the stock market goes up tomorrow is 3 5, then what are the odds a) in favor of the stock market going up tomorrow? 3 to 2 b) against the stock market going up tomorrow? 2 to If the probability of a coal miners strike this year is 0, then what are the odds a) in favor of a strike? 9 to b) against a strike? to If the odds are 3 to in favor of the Black Hawks winning their next game, then a) what are the odds against the Black Hawks winning their next game? to 3 b) what is the probability that the Black Hawks win their next game? If the odds are 5 to against the Democratic presidential nominee winning the election, then a) what are the odds in favor of the Democrat winning the election? to 5 b) what is the probability that the Democrat wins the election? What are the odds in favor of getting exactly 2 heads in 3 tosses of a coin? 3 to 5. What are the odds in favor of getting a 6 in a single toss of a die? to What are the odds in favor of getting a sum of 8 when tossing a pair of dice? 5 to What are the odds in favor of getting at least one 6 when tossing a pair of dice? to If one million lottery tickets are sold and only one of them is the winning ticket, then what are the odds in favor of winning if you hold a single ticket? to 999, What are the odds in favor of winning a lottery where you must choose 6 numbers from the numbers through 49? to 3,983,85 4. If the odds in favor of getting 5 heads in 5 tosses of a coin are to 3, then what is the probability of getting 5 heads in 5 tosses of a coin? If the odds against Smith winning the election are 2 to 5, then what is the probability that Smith wins the election? 5 7 GETTING MORE INVOLVED 43. In the Louisiana Lottery a player chooses 6 numbers from the numbers through 44. You win the big prize if the 6 chosen numbers match the 6 winning numbers chosen on Saturday night. a) What is the probability that you choose all 6 winning numbers? b) What is the probability that you do not get all 6 winning numbers? c) What are the odds in favor of winning the big prize with a single entry?, 7, 059, 05, to 7,059,05 7,059,052 7, 059, In the Louisiana Power Ball a player chooses 5 numbers from the numbers through 49 and one number (the power ball) from through 42. a) How many ways are there to choose the 5 numbers and, choose the power ball? b) What is the probability of winning the big prize in the Power Ball Lottery? c) What are the odds in favor of winning the big prize? 80,089,28,, to 80,089,27 80,08 9,28
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