Probability is the chance that some event will occur. Asimple event is one outcome or a collection ofoutcomes. What is an outcome?
Real-World Link
For a sledding trip, you randomly select one of the fourhats shown. Complete the table to show the possibleoutcomes.
1. Write a ratio that compares the number of blue hats tothe total number of hats.
2. Describe a hat display in which you would have abetter chance of selecting a red hat.
Probability
Probability
Probability can be written as a fraction, decimal, or percent.
Outcomes occur at random if each outcome is equally likelyto occur.
Example 1
Got it? 1
Example 2
Got it? 2
Find the probability of each event. Write your answer ina fractions, percent, and decimal.
a. P(F) b. P(D or G)c. P(vowel)
Find the Probability of the Complement
Complementary events are two events in which either oneor the other must happen, but cannot happen at the sametime.
For example, a coin can either land on heads, or notheads.
The sum of the probability and complement is 1 or 100%.
Example 3
Example 3
Got it? 3
Example 4
THEORETICAL ANDEXPERIMENTAL PROBABILITY
Lesson 9-2
Real-World Link
A prize wheels for a carnival game are shown. Youreceive a less expensive prize if you spin and win wheel A.You receive a more expensive prize if you spin and winwheel B.
1. Which wheel has a uniform probability?
2. Why do you think winners on wheel A receive a lessexpensive 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 probabilityexperiment.
Experimental probability is based on relative frequency –what actually occurs during an experiment.
Experimental and Theoretical Probability
The theoretical probability and the experimentalprobability of an event may or may not be the same.
As the number of attempts increases, the theoreticalprobability and the experimental probability shouldbecome closer in value.
Example 1
The graph shows the results of anexperiment in which a spinner with 3equal sections is spun sixty times. Findthe experimental probability of spinning ared for this experiment.
Example 2
Got it? 1 & 2
a. Refer to Example 1. If the spinner was spun 3more times and landed on green each time, find theexperimental probability of spinning green for thisexperiment.
b. Compare the experimental probability you foundto its theoretical probability.
Example 3
Example 4
Got it? 3 & 4
a. In Example 3, what is the experimental probability ofrolling a sum that is not 9?
b. Suppose three coins are tossed 10 times. All threecoins land on heads 1 time. Compare the experimentalprobability to the theoretical probability. If theprobabilities are not close, explain a possible reason forthe discrepancy.
Example 5 – Predict Future Events
PROBABILITY OFCOMPOUND EVENTS
Lesson 9-3
Sample Space and Tree Diagram
Sample Space
(all possible outcomes)
Sample space of rollinga die and flipping acoin.
{1H, 2H, 3H, 4H, 5H, 6H,1T, 2T, 3T, 4T, 5T, 6T}
Tree Diagram
(one way to show samplespace)
Example 1
The three students chosen to represent Mr. Balderick’sclass in a school assembly are shown. All three needto sit in a row on the stage. Find the sample space forthe different ways they can sit in a row.
Students
Adrienne
Carlos
Greg
Use A for Adrienne, C for Carolos and Gfor Greg.
ACG, AGCThe sample
CAG, CGAspace contains
GAC, GCA6 outcomes.
Example 2
A car can be purchasedin blue, silver, red, orpurple. It also comes as aconvertible or hardtop.Use a table or treediagram to fine thesample space of thedifferent colors and stylesof each car.
The sample space contains 8 outcomes.
Color
Top
Blue
Convertible
Blue
Hardtop
Silver
Convertible
Silver
Hardtop
Red
Convertible
Red
Hardtop
Purple
Convertible
Purple
Hardtop
Got it? 1 & 2
The table shows the sandwich choice for a picnic. Findthe sample space using a list, table or tree diagramfor a sandwich consisting of one type of meat andone type of bread.
HR, HS, HW, TR, TS, TW
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 ofgetting three tails?
Got it? 3
Example 4
To win a carnival prize, youneed to choose one of 3 doorslabeled 1, 2, and 3. Then youneed to choose a red, yellow,or blue box behind each door.What is the probability that theprize is in the blue or yellowbox behind door 2?
SIMULATIONS
Lesson 9-4
Model Equally Likely Outcomes
A simulation is an experiment that is designed tomodel the action in a given situation. For example,you use a random number generator to simulaterolling a dice.
Simulations often use models to act out an event thatwould be impractical to perform.
Example 1
A cereal company is placing one of eight different tradingcards in its boxes of cereal. If each card is equally likely toappear in a box of cereal, describe a model that could be usedto 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.
Example 2
Every student who volunteers at theconcession stand during basketball gameswill receive a free school T-shirt. The T-shirts come in three different designs.
Design a simulation that could be used tomodel this situation.
Use a spinner with three sections to represent three designs.
Based on the simulation, a student should volunteer four times in orderto get all 3 T-shirts.
Got it? 1 & 2
a. A restaurant is giving away 1 of 5 different toyswith its children’s meals. If the toys are given outrandomly, describe a model that could be used tosimulate which toys would be given with 6children’s meals.
One answer: use a spinner with 5 equal sections, spinthe spinner 6 times.
Got it? 1 & 2
b. Mr. Chen must wear a dress shirt and tie towork. Each day he picks one of his 6 ties atrandom. Design a simulation that could be used tomodel this situation.
One answer: use a single die to represent one of thesix different ties.
Example 3
Got it? 3
FUNDAMENTAL COUNTINGPRINCIPLE
Lesson 9-5
Fundamental Counting Principle
If event M has m possible outcomes and even N has npossible outcomes, then event M followed by event Nhas m x n possible outcomes.
You can use multiplication instead of making a treediagram to find the number of possible outcomes in asample space. This is called the FundamentalCounting Principle.
Example 1
Find the total number of outcomes when a coin istossed and a number cube is rolled.
A coin has 2 possible outcomes and a die has 6 possibleoutcomes. Multiple the possible outcomes together.
There are 42 possible outcomes.
Got it? 1
Find the total number of outcomes when choosingfrom bike helmets that come in three colors and twostyles.
6
Example 2
Example 3
Got it? 2 & 3
Example 4
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 thenumber of permutations.
Example 1
Julia is scheduling her first three classes. Her choicesare math, science, and language arts. Find thenumber of different ways Julia can schedule her firstthree classes.
Example 2
An ice cream shop has 31 flavors. Carlos wants tobuy a three-scoop cone with three different flavors.How many cones could he buy if the order of flavorsare 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 ofa volleyball team stand in a row for a picture?
720
b. In a race with 7 runners, in how many ways canthe runners end up in first, second, and third?
210
Permutations
The symbol P(31,3) represents the number ofpermutations of 31 things taken 3 at a time.
Example 3
Find P(8, 3).
P(8, 3) = 8 • 7 • 6
= 336
Example 4
Example 5
INDEPENDENT ANDDEPENDENT EVENTS
Lesson 9-7
Independent Events
Independent Events is when one event does notaffect another event.
Key Concept:
We will continue to use tree diagrams to show samplespace.
Example 1
One letter tile is selected and the spinner is spun. Whatis the probability that both will be a vowel?
Make a tree diagram
Example 1
One letter tile is selected and the spinner is spun. Whatis the probability that both will be a vowel?
Use Multiplication
Example 2
Probability of Dependent Events
If the outcome of one event affects another event, theevents are dependent.