CC- Probability of Compound Events Common Core State Standards 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 Start with a plan How many songs are there? 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? Artist Absolute Value Algebras Arithmetics FOILs Pascal s Triangle Pi MATHEMATICAL PRACTICES 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 Lesson Vocabulary compound event mutually exclusive events overlapping events independent events dependent events 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 HSM_A //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/In
Problem Mutually Exclusive and Overlapping Events Suppose you spin a spinner that has equal-sized sections numbered from to A What is the probability that you spin a or a? Because the spinner cannot land on both and, the events are mutually exclusive P ( or ) = P () + P () = + = = Substitute Simplify 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? Since a number can be a multiple of and a multiple of, such as, the events are overlapping 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 = P (multiple of ) + P (multiple of ) - P (multiple of and ) = + - = = Substitute Simplify The probability that you spin a number that is a multiple of or a multiple of is Got It? 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? 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 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) Reg_SE_CC TrKitindd Page // : PM gg- nterior_files/a Lesson - CC- # P(B) Probability of Compound Events Probability//PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/A of Compound Events
Problem Finding the Probability of Independent Events 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? Are the events independent? Yes The outcome of rolling one number cube does not affect the outcome of rolling another number cube P (red ) = Only one of the six numbers is a Three of the six numbers are even P (blue even) = = P (red and blue even) = P (red ) = # P (blue even) # = Substitute and then simplify The probability is Got It? You roll a red number cube and a blue number cube What is the probability that you roll a on the red cube and a or on the blue cube? Problem Selecting With Replacement 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? Because you replace the first tile, the events are independent P (dotted) = of the tiles are dotted P (dragon) = = P (dotted and dragon) = P (dotted) = = # of the tiles are dragons # P (dragon) Substitute Simplify The probability that you will choose a dotted tile and then a dragon tile is Got It? In Problem, what is the probability that you randomly choose a bird and then, after replacing the first tile, a flower? 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 Chapter Data Analysis and Probability HSM_AReg_SE_CC TrKitindd // : PM gg- CommonPage Core HSM_A //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/In
Key Concept Probability of Two Dependent Events If A and B are dependent events, P (A then B) = P (A) # P (B after A) Problem Selecting Without Replacement 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 (dotted then dragon) = P (dotted) = / # P (dragon after dotted) # = Substitute and then simplify The probability that you will choose a dotted tile and then a dragon tile is Got It? 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 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 Determine whether the events are dependent or independent and use the formula that applies P (junior then senior) The first outcome affects the probability of the second So the events are dependent P (junior) = = of the students are juniors P (senior after junior) = 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 Reg_SE_CC TrKitindd Page // : PM epg nterior_files/a Lesson - CC- Probability of Compound Events Probability//PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Interior_Files/S of Compound Events
Got It? 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 Lesson Check Do you know HOW? 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 B MATHEMATICAL Do you UNDERSTAND? 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? a P(B or number) b P(red or ) c P(red or yellow) d P(yellow or letter) Open-Ended What is a real-world example of two independent events? 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? = MATHEMATICAL Practice and Problem-Solving Exercises hsmase tai PRACTICES A Practice P( or ) P(even or red) P(odd or ) 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) You roll a blue number cube and a green number cube Find each probability P(blue or and green ) P (B and B) P (C and C) See Problem P(green less than and blue ) P (A and B) P(blue and green both less than ) P (A and A) P(blue even and green even) 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 See Problem You spin the spinner at the right, which is divided into equal sections Find each probability hsmase t See Problem P (B and C) Chapter Data Analysis and Probability HSM_AReg_SE_CC TrKitindd Page // : PM gg- Common Core HSM_A //PE/TRANSITION_KITS/NA/ANCILLARY//XXXXXXXXXX/Layout/Inter
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 See Problem P(dime then nickel) P(quarter then penny) P(penny then dime) P(penny then quarter) P(penny then nickel) P(dime then penny) P(dime then dime) P(quarter then quarter) 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? See Problem B Apply 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? Are the two events dependent or independent? Explain Toss a penny Then toss a nickel 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 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? 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? 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? Percent of State That Is Farmland Alabama % Florida % Indiana % CC- Probability of Compound Events
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? Common Core