CC- Interactive Learning Solve It! PURPOSE To determine the probability of a compound event using simple probability PROCESS Students may use simple probability by determining the number of favorable outcomes and comparing it to the number of possible outcomes Common Core State Standards Probability of Compound Events MACCS-CP Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model Also MACCS-CP MP, MP, MP, MP, MP Objectives To find probabilities of mutually exclusive and overlapping events To find probabilities of independent and dependent events FACILITATE q How many songs on the music device are rock songs? The portable music player at the right is set to choose a song at random from the playlist What is the probability that the next song played is a rock song by an artist whose name begins with the letter A? How did you find your answer? Start with a plan How many songs are there? How many are performed by an artist whose name begins with the letter A? [; ] q How many songs on the music device are rock songs that are performed by an artist whose name begins with the letter A? [] Artist Absolute Value Algebras Arithmetics FOILs Pascal s Triangle Pi MATHEMATICAL PRACTICES ANSWER See Solve It in Answers on next page CONNECT THE MATH Students explore a compound Category Songs Rock Pop Rock Pop Country Rock In the Solve It, you found the probability that the next song is both a rock song and also a song by an artist whose name begins with the letter A This is an example of a compound event, which consists of two or more events linked by the word and or the word or event in the Solve It In the lesson, students will learn about compound events, mutually exclusive events, overlapping events, independent events, and dependent events and how the type of event affects the probability of the event Lesson Vocabulary compound event mutually exclusive events overlapping events independent events dependent events Guided Instruction Take Note Use Venn diagrams and the data provided in the Solve It to illustrate the concepts of mutually exclusive and overlapping events How man are there There are :,,,,, an multiple, and multiples o and Essential Understanding You can write the probability of a compound event as an expression involving probabilities of simpler events This may make the compound probability easier to find When two events have no outcomes in common, the events are mutually exclusive events If A and B are mutually exclusive events, then P(A and B) = When events have at least one outcome in common, they are overlapping events You need to determine whether two events A and B are mutually exclusive before you can find P (A or B) Key Concept Probability of A or B Probability of Mutually Exclusive Events If A and B are mutually exclusive events, P (A or B) = P(A) + P(B) Probability of Overlapping Events If A and B are overlapping events, P (A or B) = P (A) + P (B) - P (A and B) Chapter Data Analysis and Probability HSM_AReg_SE_CC TrKitindd // : PM gg- CommonPage Core CC- Preparing to Teach BIG idea Probability Essential Understandings The probability of a compound event can sometimes be found from expressions of the probabilities of simpler events Different methods must be used for finding the probability of two dependent events compared to finding the probability of two independent events Math Background A compound event in the study of probability is an event that consists of two or more simple probability events When two simple events constitute a compound event, the two events can be either a union in which one or the other event occurs, or an intersection in which both of the events occur The two events are said to be mutually exclusive if the probability of both events occurring is zero The two Common Core events are said to be independent if the probability of one event occurring is not HSM_AReg_SE_TrKitindd Page // : PM gg- dependent on the other event occurring Have students look at compound probability problems by first determining whether the first event affects the second event Once students determine whether the events are dependent or independent, they can select the appropriate equation It may sometimes be difficult to determine whether events are independent, but it is crucial mathematicaly: P (A and B) = P (A) P (B) if and only if A and B are independent events Mathematical Practice Attend to precision Students will make explicit use of the terms mutually exclusive events and overlapping events and will determine when to apply each HSM_AReg_SE_CC TrKitind //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A
Problem Mutually Exclusive and Overlapping Events Problem Suppose you spin a spinner that has equal-sized sections numbered from to A What is the probability that you spin a or a? q Is it possible for the spinner to land on both and Because the spinner cannot land on both and, the events are mutually exclusive during the same spin? Explain [No; there is only P ( or ) = P () + P () = + = = one number on each section] q If you used the formula for overlapping events to Substitute determine the probability in A, would you still arrive at the same answer? [Yes; the probability Simplify of landing on both and is zero, so you would get the same answer] The probability that you spin a or a is B What is the probability that you spin a number that is a multiple of or? q Is it possible to land on both a multiple of and a Since a number can be a multiple of and a multiple of, such as, the events are overlapping multiple of? Explain [Yes; the numbers and are multiples of both and ] P (multiple of or multiple of ) How many multiples are there? There are multiples of :,,,,,,,,, and There are multiples of :,,, and There are multiples of and : and q If you used the formula for mutually exclusive = P (multiple of ) + P (multiple of ) - P (multiple of and ) = + - = = events to determine the probability in B, would you still arrive at the same answer? [No, because Substitute you would count some of the sections twice as favorable outcomes] Simplify The probability that you spin a number that is a multiple of or a multiple of is Suppose you roll a standard number cube a What is the probability that you roll an even number or a number less than? b What is the probability that you roll a or an odd number? q Which formula should you use to compute the probability in a? b? [formula for overlapping events; formula for mutually exclusive events] A standard set of checkers has equal numbers of red and black checkers The diagram at the right shows the possible outcomes when randomly choosing a checker, putting it back, and choosing again The probability of getting a red on either choice is The first choice, or event, does not affect the second event The events are independent st Choice Red nd Choice Red Black Black Take Note Ask students to state several more examples and a nonexample of independent events Red Black Two events are independent events if the occurrence of one event does not affect the probability of the second event hsmase tai Key Concept Probability of Two Independent Events If A and B are independent events, P (A and B) = P(A) Lesson - HSM_AReg_SE_CC TrKitindd Page // : PM gg- XXXXXXX/Layout/Interior_Files/A P(B) Probability of Compound Events CC- Probability//PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A of Compound Events Answers HSM_AReg_SE_TrKitindd Page // : PM gg- //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A Solve It! ; explanations may vary a b CC-
Problem Finding the Probability of Independent Events Problem Suppose you roll a red number cube and a blue number cube What is the probability that you will roll a on the red cube and an even number on the blue cube? The probability can be calculated by interpreting this event as a simple event q What is the total number of outcomes possible when the two number cubes are rolled simultaneously? Explain [Using the Multiplication Counting Principle, there are outcomes] = possible q What is the total number of favorable outcomes P (red ) = Are the events independent? Yes The outcome of rolling one number cube does not affect the outcome of rolling another number cube Three of the six numbers are even P (red and blue even) = P (red ) = The probability is P (blue even) = Substitute and then simplify that you roll a on the red cube and a or on the blue cube? = possible Problem Selecting With Replacement Ask students to describe an event involving the number cubes that has a probability of Problem Show students that the probability can be computed using the Multiplication Counting Principle The number of ways to choose a dotted tile first and then a dragon is = ways The number of ways to choose two tiles is = Therefore, the probability is = Why are the events independent when you select with replacement? When you replace the tile, the conditions for the second selection are exactly the same as for the first selection Games You choose a tile at random from the game tiles shown You replace the first tile and then choose again What is the probability that you choose a dotted tile and then a dragon tile? P (dotted) = of the tiles are dotted P (dragon) = = of the tiles are dragons P (dotted and dragon) = P (dotted) = = P (dragon) Substitute Simplify The probability that you will choose a dotted tile and then a dragon tile is In Problem, what is the probability that you randomly choose a bird and then, after replacing the first tile, a flower? q What is the probability of choosing a bird tile? a Two events are dependent events if the occurrence of one event affects the probability of the second event For example, suppose in Problem that you do not replace the first tile before choosing another This changes the set of possible outcomes for your second selection ; ] flower tile? [ Chapter Data Analysis and Probability HSM_AReg_SE_CC TrKitindd // : PM gg- CommonPage Core HSM_AReg_SE_CC TrKitind //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A Additional Problems HSM_AReg_SE_TrKitindd Page // : PM gg- A dartboard has equally sized sections numbered from to a What is the probability of throwing a dart that lands on an odd number? b What is the probability of throwing a dart that lands on a multiple of? Answer a b Suppose you roll a number cube and flip a coin What is the probability of rolling a number greater than and flipping heads? Answer A bag contains red chips, green chips, blue chips, and black chips Andrew selects a chip at random He replaces the chip and then selects another one at random What is the probability that he selects a red chip, then a black chip? Answer Refer to the information given in Additional Problem Suppose Andrew selects a chip at random, does not replace it, then selects another chip at random What is the probability that he selects a blue chip, then a green chip? Answer How is P( dotted) d from P(d After selec tile withou there is on to choose second ch Because you replace the first tile, the events are independent P (blue even) = = You roll a red number cube and a blue number cube What is the probability possible when the two number cubes are rolled simultaneously? [Using the Multiplication Counting Principle, there are favorable outcomes] Only one of the six numbers is a Common Core Justin has rock songs, hip hop songs, classical music songs, and country songs in a playlist on his mp player Suppose he plays songs at random from the playlist If the mp player will not play the same song twice in a row, what is the probability that he will hear a rock song followed by a country song? Answer //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A
Take Note Key Concept Probability of Two Dependent Events If A and B are dependent events, P (A then B) = P (A) P (B after A) Ask students to state several more examples and a nonexample of dependent events Problem Selecting Without Replacement Problem Games Suppose you choose a tile at random from the tiles shown in Problem Without replacing the first tile, you select a second tile What is the probability that you choose a dotted tile and then a dragon tile? Because you do not replace the first tile, the events are dependent How is P(dragon after dotted) different from P(dragon)? After selecting the first tile without replacement, there is one less tile to choose from for the second choice P (dotted) = / of the tiles are dotted P (dragon after dotted) = of the remaining tiles are dragons P (dragon after dotted) = Substitute and then simplify P (dotted then dragon) = P (dotted) = Show students that the probability can be computed using the Multiplication Counting Principle The number of ways to choose a dotted tile first and then a dragon is = ways The number of ways to choose two tiles is = Therefore, the probability is = The probability that you will choose a dotted tile and then a dragon tile is In Problem, what is the probability that you will randomly choose a flower hsmase t and then, without replacing the first tile, a bird? Problem Finding the Probability of a Compound Event name being chosen, what might occur? [One student would need to read his or her essay twice] q What is the probability of choosing both of the sophomores to read their essays? Explain [ = = ] The first outcome affects the probability of the second So the events are dependent P (junior) = = P (senior after junior) = flower and then, without replacing the first tile, another flower? [The probability is zero] q If the first name were replaced prior to the second Determine whether the events are dependent or independent and use the formula that applies P (junior then senior) q What is the probability that you will choose a Problem Essay Contest One freshman, sophomores, juniors, and seniors receive top scores in a school essay contest To choose which students will read their essays at the town fair, names are chosen at random from a hat What is the probability that a junior and then a senior are chosen? Grade levels of the students of the students are juniors of the remaining students are seniors P (junior then senior) = P (junior) P (senior after junior) = = Substitute and then simplify The probability that a junior and then a senior are chosen is Lesson - HSM_AReg_SE_CC TrKitindd Page // : PM epg XXXXXXX/Layout/Interior_Files/A Probability of Compound Events CC- Probability//PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/S of Compound Events Answers HSM_AReg_SE_TrKitindd Page // : AM epg //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/S (continued) CC-
a In Problem, what is the probability that a senior and then a junior are chosen? b Reasoning Is P(junior then senior) different from P (senior then junior)? Explain q In problem, what is the probability that no seniors ] or juniors are chosen? [ Lesson Check Lesson Check Do you know HOW? B If students have difficulty with Exercise, then have them review Problem to understand how to handle replacement D Reasoning Are an event and its complement mutually exclusive or overlapping? Use an example to explain You choose a card at random What is each probability? Do you UNDERSTAND? If students have difficulty with Exercise, then have them also provide an example of a compound event composed of two mutually exclusive events when you spin a spinner with the integers from through PRACTICES Vocabulary What is an example of a compound event composed of two overlapping events when you spin a spinner with the integers from through? Use the cards below Do you know HOW? MATHEMATICAL Do you UNDERSTAND? a P(B or number) b P(red or ) c P(red or yellow) d P(yellow or letter) B Open-Ended What is a real-world example of two independent events? App Error Analysis Describe and correct the error below in calculating P(yellow or letter) from Exercise, part (d) What is the probability of choosing a yellow card and then a D if the first card is not replaced before the second card is drawn? P(yellow or letter) = P(yellow) or P(letter) = + What is the probability of choosing a yellow card and then a D if the first card is replaced before the second card is drawn? = Close q How does finding the probability of selecting with replacement compare to finding the probability of selecting without replacement? [When you select MATHEMATICAL Practice and Problem-Solving Exercises hsmase tai PRACTICES A Practice with replacement, the total number of possible outcomes is the same for each event When you select without replacement, the total number of possible outcomes decreases after each event] See Problem You spin the spinner at the right, which is divided into equal sections Find each probability P( or ) P(even or red) P( or red) P(red or less than ) P(odd or multiple of ) P( or blue) P(red or more than ) P(greater than or blue) P(blue or and green ) P (A and B) P (B and B) hsmase tai See Problem Answers HSM_AReg_SE_TrKitindd Page // : AM epg (continued) a b No; the numerators and the denominators are the same, so the product is the same Lesson Check a b c d Answers may vary Sample: find the probability of spinning a number less than that is even Mutually exclusive; answers may vary Sample: The complement of being even on a number die is being odd, and even and odd are mutually exclusive Common Core P (C and C) P (B and C) Data Analysis and Probability See Problem P(green less than and blue ) HSM_AReg_SE_CC TrKitindd Page // : PM gg- Common Core P(blue and green both less than ) You choose a tile at random from a bag containing A s, B s, and C s You replace the first tile in the bag and then choose again Find each probability Chapter P(blue even and green even) P (A and A) You roll a blue number cube and a green number cube Find each probability P(odd or ) HSM_AReg_SE_CC TrKitind //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A Check students work //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/S Because a tile can be both yellow and a letter, the formula should be P (yellow or letter) = P (yellow) + P (letter) - P (yellow and letter) = + - = Practice and Problem-Solving Exercises
You pick a coin at random from the set shown at the right and then pick a second coin without replacing the first Find each probability P(dime then nickel) See Problem P(quarter then penny) Practice P(penny then dime) P(penny then quarter) ASSIGNMENT GUIDE P(penny then nickel) P(dime then penny) Basic: all, even, P(dime then dime) P(quarter then quarter) Average: odd, Cafeteria Each day, you, Terry, and other friends randomly choose one of your names from a hat to decide who throws away everyone s lunch trash What is the probability that you are chosen on Monday and Terry is chosen on Tuesday? u Advanced: odd, See Problem Mathematical Practices are supported by exercises with red headings Here are the Practices supported in this lesson: Free Samples Samples of a new drink are handed out at random from a cooler holding citrus drinks, apple drinks, and raspberry drinks What is the probability that an apple drink and then a citrus drink are handed out? lly B Apply MP : Make Sense of Problems Ex MP : Reason Abstractly Ex MP : Reason Quantitatively Ex MP : Communicate Ex MP : Critique the Reasoning of Others Ex MP : Model with Mathematics Ex, Are the two events dependent or independent? Explain Toss a penny Then toss a nickel in ) Pick a name from a hat Without replacement, pick a different name Pick a ball from a basket of yellow and pink balls Return the ball and pick again Writing Use your own words to explain the difference between independent and dependent events Give an example of each Applications exercises have blue headings Exercise supports MP : Model Reasoning A bag holds yellow mints and other green or pink mints You choose a mint at random, eat it, and choose another a Find the number of pink mints if P (yellow then pink) = P (green then yellow) b What is the least number of pink mints if P (yellow then pink) P (green then yellow)? Think About a Plan An acre of land is chosen at random from each of the three states listed in the table at the right What is the probability that all three acres will be farmland? Does the choice of an acre from one state affect the choice from the other states? How must you rewrite the percents to use a formula from this lesson? em HOMEWORK QUICK CHECK To check students understanding of key skills and concepts, go over Exercises,,,, and Percent of State That Is Farmland Alabama % Florida % Indiana % Phone Poll A pollster conducts a survey by phone The probability that a call does not result in a person taking this survey is % What is the probability that the pollster makes calls and none result in a person taking the survey? em Open-Ended Find the number of left-handed students and the number of righthanded students in your class Suppose your teacher randomly selects one student to take attendance and then a different student to work on a problem on the board a What is the probability that both students are left-handed? b What is the probability that both students are right-handed? c What is the probability that the first student is right-handed and the second student is left-handed? _tai em Lesson - HSM_AReg_SE_TrKitindd Page // : PM gg- CC- Probability//PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A of Compound Events HSM_AReg_SE_CC TrKitindd Page // : PM gg- XXXXXXX/Layout/Interior_Files/A Probability of Compound Events Independent; the outcome of the first event does not affect the second event Dependent; the outcome of the first event affects the outcome of the second Independent; the outcome of the first event does not affect the second event For independent events, the outcome of the first event does not affect the outcome of the second event, while for dependent events, the outcome is affected An example of two independent events is the rolling of two number cubes An example of two dependent events is picking two cards from a deck without replacing the first one a pink mints b pink mints about % about % a c Check students work //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A CC-
Answers Practice and Problem-Solving Exercises (continued) a b c a C Challenge Suppose you roll a red number cube and a yellow number cube a What is P(red and yellow )? b What is P(red and yellow )? c What is the probability of rolling any matching pair of numbers? (Hint: Add the probabilities of each of the six matches) A two-digit number is formed by randomly selecting from the digits,,, and without replacement a How many different two-digit numbers can be formed? b What is the probability that a two-digit number contains a or a? c What is the probability that a two-digit number is prime? b c Common Core Common Core