Probability of Simple Events Lesson 9-1

Vocabulary Start-Up Probability is the chance that some event will occur. A simple event is one outcome or a collection of outcomes. What is an outcome?

Real-World Link For a sledding trip, you randomly select one of the four hats shown. Complete the table to show the possible outcomes. 1. Write a ratio that compares the number of blue hats to the total number of hats. 2. Describe a hat display in which you would have a better chance of selecting a red hat. 1:4 or 1 4

Probability Words: The probability of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. Symbols: P(event) = number of favorable events number of possible outcomes

Probability Probability can be written as a fraction, decimal, or percent. Outcomes occur at random if each outcome is equally likely to occur.

Example 1 There are six equally likely outcomes if a die with six sides labeled 1 through 6 is rolled. Find the P(6) of the probability of rolling a 6. P(6) = number of favorable events number of possible outcomes = 1 6 The probability of rolling a 6 is 1 6 , or about 17%, or about 0.17.

Got it? 1 A coin is tossed. Find the probability of the coin landing on heads. Write your answer as a fraction, percent, and decimal. P(6) = number of favorable events number of possible outcomes = 1 2 The probability of landing on heads is 1 5 , or 50%, or 0.5.

Example 2 Find the probability of rolling a 2, 3, or 4 on a die. P(2, 3, or 4) = number of favorable events number of possible outcomes = 3 6 = 1 2 The probability of rolling a 2, 3, or 4 is 1 2 , or 50%, or 0.5.

Got it? 2 Find the probability of each event. Write your answer in a fractions, percent, and decimal. a. P(F) b. P(D or G) c. P(vowel) 1 10 , 10%, 0.1 1 5 , 20%, 0.2 3 10 , 30%, 0.3

Find the Probability of the Complement Complementary events are two events in which either one or the other must happen, but cannot happen at the same time. For example, a coin can either land on heads, or not heads. The sum of the probability and complement is 1 or 100%.

Example 3 Find the probability of not rolling a 6 in Example 1. Method 1: The probability of not rolling a 6 and rolling a 6 are complimentary, so the sum or the probabilities is 1. P(6) + P(not 6) = 1 1 6 + P(not 6) = 1 The probability of not rolling a 6 is 5 6 .

Example 3 Find the probability of not rolling a 6 in Example 1. Method 2: Think: How many “not sixes” are on the die? 5 So, the probability is 5 6 . The probability of not rolling a 6 is 5 6 .

Got it? 3 A bag contains 5 blue, 8 red, and 7 green marbles. A marble is selected at random. Find the probability the marble is not red. 3 5 , 60%, 0.6

Example 4 Mr. Haranda surveyed his class and discovered that 30% of his students have blue eyes. Identify the complement of this event. Then find the probability. Think: The compliment of having blue eyes is not have blue eyes. 30% + P(not blue eyes) = 100% 30% + 70% = 100% The probability of not having blue eyes is 70%, 0.7, or 7 10 .

Theoretical and Experimental Probability Lesson 9-2

Real-World Link A prize wheels for a carnival game are shown. You receive a less expensive prize if you spin and win wheel A. You receive a more expensive prize if you spin and win wheel B. 1. Which wheel has a uniform probability? 2. Why do you think winners on wheel A receive a less expensive prize than winners on wheel B? Wheel A

Experimental and Theoretical Probability Theoretical probability is based on uniform probability – what should happen when conducting a probability experiment. Experimental probability is based on relative frequency – what actually occurs during an experiment.

Experimental and Theoretical Probability The theoretical probability and the experimental probability of an event may or may not be the same. As the number of attempts increases, the theoretical probability and the experimental probability should become closer in value.

Example 1 The graph shows the results of an experiment in which a spinner with 3 equal sections is spun sixty times. Find the experimental probability of spinning a red for this experiment. The graph indicates that the spinner landed on red 24 times, blue 15 times, and green 21 times. P(red) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑟𝑒𝑑 𝑜𝑐𝑐𝑢𝑟𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑝𝑖𝑛𝑠 = 24 60 or 2 5 The experimental probability of spinning red is 2 5 .

Example 2 Compare the experimental probability you found in Example 1 to its theoretical probability. The spinner has three equal sections: red, blue, and green. So the theoretical probability of spinning red is 1 3 . Since 2 5 ≈ 1 3 , the experimental probability is close to the theoretical probability.

Got it? 1 & 2 a. Refer to Example 1. If the spinner was spun 3 more times and landed on green each time, find the experimental probability of spinning green for this experiment. b. Compare the experimental probability you found to its theoretical probability. 8 21 The experiemental probability is close to the theoretical probability since 8 21 ≈ 1 3 .

Example 3 Two dice are rolled together 20 times. A sum of 9 is rolled 8 times. What is the experimental probability of rolling a sum of 9? P(9) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎 𝑠𝑢𝑚 𝑜𝑓 9 𝑜𝑐𝑐𝑢𝑟𝑠 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑜𝑙𝑙𝑠 = 8 20 or 2 5 The experimental probability of rolling a sum of 9 is 2 5 .

Example 4 Compare the experimental probability you found in Example 3 to its theoretical probability. If the probabilities are not close, explain a possible reason for the discrepancy. When rolling two dice, there are 36 possibilities, the theoretical probability is 4 36 or 1 9 . The theoretical probability to not close to the experimental probability. One possible explanation is that is not enough trials.

Got it? 3 & 4 a. In Example 3, what is the experimental probability of rolling a sum that is not 9? b. Suppose three coins are tossed 10 times. All three coins land on heads 1 time. Compare the experimental probability to the theoretical probability. If the probabilities are not close, explain a possible reason for the discrepancy. 3 5 1 10 is close to 1 8 .

Example 5 – Predict Future Events Last year, a DVD store sold 670 action DVDs, 580 comedy DVDs, 450 drama DVDs and 300 horror DVDs. A media buyer expects to sell 5,000 DVDs this year. Based on these results, how many comedy DVDs should she buy? Explain. 2,000 DVDs were sold and 580 were comedy. The probability is 580 2000 or 29 100 . 29 100 = 𝑥 5000 29(5,000) = 100x 145,000 = 100x 1,450 = x She should buy about 1,450 comedy DVDs.

Probability of compound events Lesson 9-3

Sample Space and Tree Diagram Sample Space (all possible outcomes) Sample space of rolling a die and flipping a coin. {1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T} Tree Diagram (one way to show sample space)

Example 1 The three students chosen to represent Mr. Balderick’s class in a school assembly are shown. All three need to sit in a row on the stage. Find the sample space for the different ways they can sit in a row. Students Adrienne Carlos Greg Use A for Adrienne, C for Carolos and G for Greg. ACG, AGC The sample CAG, CGA space contains GAC, GCA 6 outcomes.

Example 2 A car can be purchased in blue, silver, red, or purple. It also comes as a convertible or hardtop. Use a table or tree diagram to fine the sample space of the different colors and styles of each car. Color Top Blue Convertible Hardtop Silver Red Purple The sample space contains 8 outcomes.

Got it? 1 & 2 The table shows the sandwich choice 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. HR, HS, HW, TR, TS, TW

The probability of getting three tails is 1 8 . Example 3 Compound Event = two or more events Suppose you toss a quarter, a dime, and a nickel. Find the sample space. What is the probability of getting three tails? The probability of getting three tails is 1 8 .

Got it? 3 The animal shelter has both male and female Labrador Retrievers in yellow, born, or black. There is an equal number of each kind. What is the probability of choosing a female yellow Labrador Retriever? P(female yellow lab) = 1 6

The probability of getting a blue or yellow box behind door 2 is 2 9 . Example 4 To win a carnival prize, you need to choose one of 3 doors labeled 1, 2, and 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? The probability of getting a blue or yellow box behind door 2 is 2 9 .

Simulations Lesson 9-4

Model Equally Likely Outcomes A simulation is an experiment that is designed to model the action in a given situation. For example, you use a random number generator to simulate rolling a dice. Simulations often use models to act out an event that would be impractical to perform.

Example 1 A cereal company is placing one of eight different trading cards in its boxes of cereal. If each card is equally likely to appear in a box of cereal, describe a model that could be used to simulate the cards you would find in 15 boxes of cereal. Choose a method that has 8 different outcome. One way is with three coins.

Use a spinner with three sections to represent three designs. Example 2 Every student who volunteers at the concession stand during basketball games will receive a free school T-shirt. The T- shirts come in three different designs. Design a simulation that could be used to model this situation. Use a spinner with three sections to represent three designs. Based on the simulation, a student should volunteer four times in order to get all 3 T-shirts.

Got it? 1 & 2 a. A restaurant is giving away 1 of 5 different toys with its children’s meals. If the toys are given out randomly, describe a model that could be used to simulate which toys would be given with 6 children’s meals. One answer: use a spinner with 5 equal sections, spin the spinner 6 times.

Got it? 1 & 2 b. Mr. Chen must wear a dress shirt and tie to work. Each day he picks one of his 6 ties at random. Design a simulation that could be used to model this situation. One answer: use a single die to represent one of the six different ties.

Example 3 There is a 60% chance of rain for each of the next two days. Describe a method you could use to find the experimental probability of having rain on both of the next two days. 60% = 60 100 or 3 5 Use 5 marbles: 2 red and 3 blue. The blue represents rain and the red represents no rain. Draw a marble, put it back and draw a second marble to represent two days. How could you represent a 20% chance of rain with marbles? Use 1 blue and 4 red.

Got it? 3 During the regular season, Jason made 80% of his free throws. Describe an experiment to find the experimental probability of Jason making his next two free throws. One answer: use a spinner with five sections, 4 represents making the free throw. Spin the spinner twice. (80% = 80 100 = 4 5 )

Fundamental Counting Principle Lesson 9-5

Fundamental Counting Principle If event M has m possible outcomes and even N has n possible outcomes, then event M followed by event N has m x n possible outcomes. You can use multiplication instead of making a tree diagram to find the number of possible outcomes in a sample space. This is called the Fundamental Counting Principle.

Example 1 Find the total number of outcomes when a coin is tossed and a number cube is rolled. A coin has 2 possible outcomes and a die has 6 possible outcomes. Multiple the possible outcomes together. There are 42 possible outcomes.

Got it? 1 Find the total number of outcomes when choosing from bike helmets that come in three colors and two styles. 6

Example 2 You can use the Fundamental Counting Principle to help find the probability of events. Find the total number of outcomes from rolling a die and choosing a letter in the word NUMBERS. Then find the probability of rolling a 6 and choosing an M. There are 42 different outcomes. So the probability is 1 42 or about 2%

Example 3 Find the number of different jeans available at The Jean Shop. Then find the probability of randomly selecting a size 32 x 24 slim fit. Is it likely or unlikely that the jeans would be chosen? There are 45 different types, so there’s a 1 45 or about 2%.

Got it? 2 & 3 Two dice are rolled. What is the probability that the sum of the numbers on the cube is 12? 1 36 , or about 3% How likely is it that the sum would be 12? Very unlikely

Example 4 A box of toy cars contains blue, orange, yellow, red, and black cars. A separate box contains a math and female action figure. What is the probability of randomly choosing an orange car and a female action figure? Is it likely or unlikely that this combination is chosen? There are 5 choices and 2 genders. 5 x 2 = 10 Probability = 1 10 or 10% P(orange, female) is very unlikely.

Permutations Lesson 9-6

Permutations 1. An arrangement, or listing, of objects 2. Order matters Example: Blue, Red, Green ≠ Red, Green, Blue Use the Probability Multiplication Rule to find the number of permutations.

Example 1 Julia is scheduling her first three classes. Her choices are math, science, and language arts. Find the number of different ways Julia can schedule her first three classes.

Example 2 An ice cream shop has 31 flavors. Carlos wants to buy a three-scoop cone with three different flavors. How many cones could he buy if the order of flavors are important? 31 • 30 • 29 = 26,970 He could buy 26,970 different ice cream cones.

Got it? a. In how many ways can the starting six players of a volleyball team stand in a row for a picture? 720 b. In a race with 7 runners, in how many ways can the runners end up in first, second, and third? 210

Permutations The symbol P(31,3) represents the number of permutations of 31 things taken 3 at a time.

Example 3 Find P(8, 3). P(8, 3) = 8 • 7 • 6 = 336

Example 4 Ashley’s iPod has a setting that allows the songs to play in a random order She has a playlist that contains 10 songs. What is the probability that the iPod will randomly play the first three songs in order. Find P(10, 3). P(10, 3) = 10 • 9 • 8 = 720 So the probability is 1 720 .

Example 5 A swimming event features 8 swimmers. If each swimmer has an equally likely chance of finishing in the top two, what I the probability that Yumli will be in first place and Paquita is in second place? Find the permutation of 8 things taken two at a time. P(8, 2) = 8 • 7 = 56 The probability is 1 56 .

Independent and Dependent Events Lesson 9-7

Independent Events is when one event does not affect another event.

We will continue to use tree diagrams to show sample space. Key Concept: We will continue to use tree diagrams to show sample space.

There are 12 outcomes. Two only contains only vowels. Example 1 One letter tile is selected and the spinner is spun. What is the probability that both will be a vowel? Make a tree diagram There are 12 outcomes. Two only contains only vowels. 2 12 = 1 6

P(selecting a vowel) = 2 4 𝑜𝑟 1 2 Example 1 One letter tile is selected and the spinner is spun. What is the probability that both will be a vowel? Use Multiplication P(selecting a vowel) = 2 4 𝑜𝑟 1 2 P(spinning a vowel) = 1 3 P(both vowels) = 1 2 • 1 3 = 1 6

Example 2 The spinner and dice shown are used in a game. What is the probability of a player not spinning a blue and then rolling a 3 or 4? P(not blue) = 4 5 P(3 or 4) = 2 6 or 1 3 P(not blue and 3 or 4) = 4 5 • 1 3 = 4 15

Probability of Dependent Events If the outcome of one event affects another event, the events are dependent.

Example 3 There are 4 oranges, 7 bananas, and 5 apples in a fruit basket. Ignacio selects a piece of fruit at random. Find the probability that two apples are chosen. P(first is an apple) = 5 16 P(second is an apple = 4 15 P(both are apples) = 5 16 • 4 15 = 20 240 or 1 12 The probability is 1 12 .

Got it? There are 4 oranges, 7 bananas, and 5 apples in a fruit basket. a. Find P(two bananas) 7 40 b. Find P(orange then apple) 1 12