Probability of Compound Events Lesson 3
Objective Students will be able to find probabilities of compound events using organized lists, tables, and tree diagrams. They will also understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
Real World Link Aimee wants to pack enough items to create 6 different outfits. She packs 1 jacket, 3 shirts, and 2 pairs of jeans. Can Aimee create 6 different outfits?
1. Complete the table below. Outfit Clothing Items 1 jacket, shirt 1, jeans 1 2 jacket, shirt 1, jeans 2 3
2. The table is an example of an organized list. What is another way to show the different outfits that Aimee can create? 3. Describe another situation for which you might want to list all of the possible outcomes.
Find a Sample Space The set of all of the possible outcomes in a probability experiment is called the sample space. Organized lists, tables, and tree diagrams can be used to represent a sample space. A tree diagram is a diagram used to show the sample space.
Examples 1. The three students chosen to represent Mr. Balderick s class in a school assembly are shown. All three of them need to sit in a row on the stage. Use a list to find the sample space for the different ways they can sit in a row. Students Adrienne Carlos Greg
Use A for Adrienne, C for Carlos, and G for Greg. Use each letter exactly once. ACG CGA AGC CAG GAC GCA So, the sample space consists of 6 outcomes.
2. A car can be purchased in blue, silver, red, or purple. It also comes as a convertible or hardtop. Use a table or a tree diagram to find the sample space for the different styles in which the car can be purchased.
Color blue blue silver silver red red purple purple Top convertible hardtop convertible hardtop convertible hardtop convertible hardtop
Color Top Sample Space Blue Silver Red Purple Convertible Hardtop Convertible Hardtop Convertible Hardtop Convertible Hardtop BC BH SC SH RC RH PC PH
Using either method, the sample space consists of 8 outcomes.
You try! a. The table shows the sandwich choices for a picnic. Find the sample space using a list, table, or tree diagram for a sandwich consisting of one type of meat and one type of bread. Meat ham turkey Bread rye sourdough white
List Table Tree Diagram HR Meat Bread Meat Bread HS HW TR TS TW Ham Ham Ham Turkey Turkey Turkey Rye Sourdough White Rye Sourdough White Ham Turkey Rye Sourdough White Rye Sourdough White
Find Probability A compound event consists of two or more simple events. The probability of a compound event, just as with simple events, is the fraction of outcomes in the sample space for which the compound event occurs.
Example 3. Suppose you toss a quarter, a dime, and a nickel. Find the sample space. What is the probability of getting three tails? Make a tree diagram to show the sample space.
Quarter Dime Nickel Sample Space heads heads heads tails tails heads tails heads heads tails tails tails heads tails heads, heads, heads heads, heads, tails heads, tails, heads heads, tails, tails tails, heads, heads tails, heads, tails tails, tails, heads tails, tails, tails
P(3 tails) = 1 8 number of favorable outcomes number of possible outcomes So, the probability of getting three tails is. 1 8
You try! b. The animal shelter has both male and female Labrador Retrievers in yellow, brown, or black. There is an equal number of each kind. What is the probability of choosing a female yellow Labrador Retriever? Show your work.
Lab Coat Sample Space male female yellow brown black yellow brown black male, yellow male, brown male, black female, yellow female, brown female, black
Real World Example 4. To win a carnival prize, you need to choose one of 3 doors labeled 1 through 3. Then you need to choose a red, yellow, or blue box behind each door. What is the probability that the prize is in the blue or yellow box behind door 2? Create a table to show the outcomes.
door 1 door 1 door 1 door 2 door 2 door 2 door 3 door 3 door 3 Outcomes red box yellow box blue box red box yellow box blue box red box yellow box blue box The table shows that there are 9 total outcomes. Two of the outcomes are favorable. So, the probability that the prize is in a blue or yellow box behind door 2 is 2/9.
Independent and Dependent Events Compound events are independent events if the occurrence of one does not affect the likelihood that the other will occur. If A and B are independent events, then P(A and B) = P(A) x P(B)
Dependent events also affect the likelihood of each other. If A and B are dependent events, then P(A, then B) = P(A) x P(B after A)
Example 5. A box holds the following letters: B L U E Mia picks a card at random and does not replace it in the box. Without looking, she picks another card at random. What is the probability that Mia will pick a consonant and then a vowel?
Identify the events: A = picking a consonant B = picking a vowel The compound event is dependent. P(A, then B) = P(A) x P(B after A) P(A) = P(consonant) = 2/4 P(B) = P (vowel) = 2/3
2 x 2 4 3 = 4 12 = 1 3 The probability of picking a consonant and then a vowel is ⅓. Check B U L U E B L E L E B U E B L U