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Pre-Algebra
9-1 Probability
Pledge & Moment of Silence
Pre-Algebra
9-1 Probability
Pre-Algebra HOMEWORK
Page 449 #1-8
&
Page 453 #1-6
Our Learning GoalStudents will be able to find theoretical probabilities, including dependent and independent events; estimate probabilities using experiments and simulations; use The Fundamental Counting Principle, permutations, and combinations; and convert between probability and odds of a specified outcome.
Our Learning Goal Assignments
• Learn to find he probability of an event by using the definition of
probability (9-1)
• Learn to estimate probability using experimental methods (9-2)
Pre-Algebra
9-1 Probability
Student Learning Goal Chart
Pre-Algebra
9-1 Probability
9-1 AND 9-2
FAST TRACK!
Pre-Algebra
9-1 Probability
Today’s Learning Goal Assignment
Learn to find the probability of an event by using the definition of probability.
Pre-Algebra
9-1 Probability
Lesson QuizUse the table to find the probability of each event.
1. 1 or 2 occurring
2. 3 not occurring
3. 2, 3, or 4 occurring0.874
0.351
0.794
9-2 Experimental ProbabilityToday’s Learning Goal Assignment
Learn to estimate probability using experimental methods.
9-2 Experimental Probability
Lesson Quiz: Part 11. Of 425, 234 seniors were enrolled in a math
course. Estimate the probability that a randomly selected senior is enrolled in a math course.
2. Mason made a hit 34 out of his last 125 times at bat. Estimate the probability that he will make a hit his next time at bat.
0.27, or 27%
0.55, or 55%
9-2 Experimental Probability
Lesson Quiz: Part 2
3. Christina polled 176 students about their favorite ice cream flavor. 63 students’ favorite flavor is vanilla and 40 students’ favorite flavor is strawberry. Compare the probability of a student’s liking vanilla to a student’s liking strawberry.
about 36% to about 23%
Pre-Algebra
9-1 Probability
Vocabulary
experimenttrialoutcomesample spaceeventprobabilityimpossiblecertain
Pre-Algebra
9-1 Probability
An experiment is an activity in which results are observed. Each observation is called a trial, and each result is called an outcome. The sample space is the set of all possible outcomes of an experiment.
Experiment Sample Space
flipping a coin heads, tails
rolling a number cube 1, 2, 3, 4, 5, 6
guessing the number of whole numbers jelly beans in a jar
Pre-Algebra
9-1 Probability
An event is any set of one or more outcomes. The probability of an event, written P(event), is a number from 0 (or 0%) to 1 (or 100%) that tells you how likely the event is to happen.
• A probability of 0 means the event is impossible, or can never happen.
• A probability of 1 means the event is certain, or has to happen.
• The probabilities of all the outcomes in the sample space add up to 1.
Pre-Algebra
9-1 Probability
0 0.25 0.5 0.75 1
0% 25% 50% 75% 100%
Never Happens about Alwayshappens half the time happens
14
12
340 1
Pre-Algebra
9-1 Probability
Give the probability for each outcome.
Additional Example 1A: Finding Probabilities of Outcomes in a Sample Space
A. The basketball team has a 70% chance of winning.
The probability of winning is P(win) = 70% = 0.7. The probabilities must add to 1, so the probability of not winning is P(lose) = 1 – 0.7 = 0.3, or 30%.
Pre-Algebra
9-1 Probability
Give the probability for each outcome.
Try This: Example 1A
A. The polo team has a 50% chance of winning.
The probability of winning is P(win) = 50% = 0.5. The probabilities must add to 1, so the probability of not winning is P(lose) = 1 – 0.5 = 0.5, or 50%.
Pre-Algebra
9-1 Probability
Give the probability for each outcome.
Additional Example 1B: Finding Probabilities of Outcomes in a Sample Space
B.
Three of the eight sections of the spinner are labeled 1, so a reasonable estimate of the probability that the spinner will land on 1 is
P(1) = .38
Pre-Algebra
9-1 Probability
Additional Example 1B Continued
Three of the eight sections of the spinner are labeled 2, so a reasonable estimate of the probability that the spinner will land on 2 is P(2) = .3
8
Two of the eight sections of the spinner are labeled 3, so a reasonable estimate of the probability that the spinner will land on 3 is P(3) = = .2
814
Check The probabilities of all the outcomes must add to 1.
38
38
28
++ = 1
Pre-Algebra
9-1 Probability
Give the probability for each outcome.
Try This: Example 1B
B. Rolling a number cube.
One of the six sides of a cube is labeled 1, so a reasonable estimate of the probability that the spinner will land on 1 is P(1) = . 1
6
Outcome 1 2 3 4 5 6
Probability
One of the six sides of a cube is labeled 2, so a reasonable estimate of the probability that the spinner will land on 1 is P(2) = . 1
6
Pre-Algebra
9-1 Probability
Try This: Example 1B Continued
One of the six sides of a cube is labeled 3, so a reasonable estimate of the probability that the spinner will land on 1 is P(3) = . 1
6
One of the six sides of a cube is labeled 4, so a reasonable estimate of the probability that the spinner will land on 1 is P(4) = . 1
6
One of the six sides of a cube is labeled 5, so a reasonable estimate of the probability that the spinner will land on 1 is P(5) = . 1
6
Pre-Algebra
9-1 Probability
Try This: Example 1B Continued
One of the six sides of a cube is labeled 6, so a reasonable estimate of the probability that the spinner will land on 1 is P(6) = . 1
6
Check The probabilities of all the outcomes must add to 1.
16
16
16
++ = 116
+16
+16
+
Pre-Algebra
9-1 Probability
To find the probability of an event, add the probabilities of all the outcomes included in the event.
Pre-Algebra
9-1 Probability
A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score.
Additional Example 2A: Finding Probabilities of Events
A. What is the probability of not guessing 3 or more correct?
The event “not three or more correct” consists of the outcomes 0, 1, and 2.
P(not 3 or more) = 0.031 + 0.156 + 0.313 = 0.5, or 50%.
Pre-Algebra
9-1 Probability
A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score.
Try This: Example 2A
A. What is the probability of guessing 3 or more correct?
The event “three or more correct” consists of the outcomes 3, 4, and 5.
P(3 or more) = 0.313 + 0.156 + 0.031 = 0.5, or 50%.
Pre-Algebra
9-1 Probability
A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score.
B. What is the probability of guessing between 2 and 5?
The event “between 2 and 5” consists of the outcomes 3 and 4.
P(between 2 and 5) = 0.313 + 0.156 = 0.469, or 46.9%
Additional Example 2B: Finding Probabilities of Events
Pre-Algebra
9-1 Probability
A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score.
B. What is the probability of guessing fewer than 3 correct?
The event “fewer than 3” consists of the outcomes 0, 1, and 2.
P(fewer than 3) = 0.031 + 0.156 + 0.313 = 0.5, or 50%
Try This: Example 2B
Pre-Algebra
9-1 Probability
A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score.
C. What is the probability of guessing an even number of questions correctly (not counting zero)?The event “even number correct” consists of the outcomes 2 and 4.
P(even number correct) = 0.313 + 0.156 = 0.469, or 46.9%
Additional Example 2C: Finding Probabilities of Events
Pre-Algebra
9-1 Probability
A quiz contains 5 true or false questions. Suppose you guess randomly on every question. The table below gives the probability of each score.
C. What is the probability of passing the quiz (getting 4 or 5 correct) by guessing?
The event “passing the quiz” consists of the outcomes 4 and 5.
P(passing the quiz) = 0.156 + 0.031 = 0.187, or 18.7%
Try This: Example 2C
Pre-Algebra
9-1 Probability
Additional Example 3: Problem Solving Application
Six students are in a race. Ken’s probability of winning is 0.2. Lee is twice as likely to win as Ken. Roy is as likely to win as Lee. Tracy, James, and Kadeem all have the same chance of winning. Create a table of probabilities for the sample space.
14
Pre-Algebra
9-1 Probability
Additional Example 3 Continued
11 Understand the Problem
The answer will be a table of probabilities. Each probability will be a number from 0 to 1. The probabilities of all outcomes add to 1.
List the important information:
• P(Ken) = 0.2
• P(Lee) = 2 P(Ken) = 2 0.2 = 0.4
• P(Tracy) = P(James) = P(Kadeem)
• P(Roy) = P(Lee) = 0.4 = 0.1 14
14
Pre-Algebra
9-1 Probability
Additional Example 3 Continued
22 Make a Plan
You know the probabilities add to 1, so use the strategy write an equation. Let p represent the probability for Tracy, James, and Kadeem.
P(Ken) + P(Lee) + P(Roy) + P(Tracy) + P(James) + P(Kadeem) = 1
0.2 + 0.4 + 0.1 + p + p + p = 1
0.7 + 3p = 1
Pre-Algebra
9-1 Probability
Solve33
0.7 + 3p = 1
–0.7 –0.7 Subtract 0.7 from both sides.
3p = 0.3
3p3
0.33
= Divide both sides by 3.
Additional Example 3 Continued
p = 0.1
Pre-Algebra
9-1 Probability
Look Back44
Check that the probabilities add to 1.
0.2 + 0.4 + 0.1 + 0.1 + 0.1 + 0.1 = 1
Additional Example 3 Continued
Pre-Algebra
9-1 Probability
Four students are in the Spelling Bee. Fred’s probability of winning is 0.6. Willa’s chances are one-third of Fred’s. Betty’s and Barrie’s chances are the same. Create a table of probabilities for the sample space.
Try This: Example 3
Pre-Algebra
9-1 Probability
Try This: Example 3 Continued
11 Understand the Problem
The answer will be a table of probabilities. Each probability will be a number from 0 to 1. The probabilities of all outcomes add to 1.
List the important information:
• P(Fred) = 0.6
• P(Betty) = P(Barrie)
• P(Willa) = P(Fred) = 0.6 = 0.213
13
Pre-Algebra
9-1 Probability
Try This: Example 3 Continued
22 Make a Plan
You know the probabilities add to 1, so use the strategy write an equation. Let p represent the probability for Betty and Barrie.
P(Fred) + P(Willa) + P(Betty) + P(Barrie) = 1
0.6 + 0.2 + p + p = 1
0.8 + 2p = 1
Pre-Algebra
9-1 Probability
Solve33
0.8 + 2p = 1
–0.8 –0.8 Subtract 0.8 from both sides.
2p = 0.2
Try This: Example 3 Continued
Outcome Fred Willa Betty Barrie
Probability 0.6 0.2 0.1 0.1
p = 0.1
Pre-Algebra
9-1 Probability
Look Back44
Check that the probabilities add to 1.
0.6 + 0.2 + 0.1 + 0.1 = 1
Try This: Example 3 Continued
Pre-Algebra
9-1 Probability
Lesson QuizUse the table to find the probability of each event.
1. 1 or 2 occurring
2. 3 not occurring
3. 2, 3, or 4 occurring0.874
0.351
0.794
Pre-Algebra
9-1 Probability9-2 Experimental Probability
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Warm UpUse the table to find the probability of each event.
1. A or B occurring
2. C not occurring
3. A, D, or E occurring
0.494
0.742
0.588
Pre-Algebra
9-2 Experimental Probability
Problem of the Day
A spinner has 4 colors: red, blue, yellow, and green. The green and yellow sections are equal in size. If the probability of not spinning red or blue is 40%, what is the probability of spinning green? 20%
Today’s Learning Goal Assignment
Learn to estimate probability using experimental methods.
Vocabulary
experimental probability
In experimental probability, the likelihood of an event is estimated by repeating an experiment many times and observing the number of times the event happens. That number is divided by the total number of trials. The more the experiment is repeated, the more accurate the estimate is likely to be.
number of times the event occurs
total number of trialsprobability
A. The table shows the results of 500 spins of a spinner. Estimate the probability of the spinner landing on 2.
Additional Example 1A: Estimating the Probability of an Event
The probability of landing on 2 is about 0.372, or 37.2%.
probability 500186number of spins that landed on 2
total number of spins =
Try This: Example 1A
A. Jeff tosses a quarter 1000 times and finds that it lands heads 523 times. What is the probability that the next toss will land heads? Tails?
P(heads) =
P(heads) + P(tails) = 1 The probabilities must equal 1.
0.523 + P(tails) = 1
P(tails) = 0.477
= 0.5231000523
B. A customs officer at the New York–Canada border noticed that of the 60 cars that he saw, 28 had New York license plates, 21 had Canadian license plates, and 11 had other license plates. Estimate the probability that a car will have Canadian license plates.
= 0.35The probability that a car will have Canadian license plates is about 0.35, or 35%.
Additional Example 1B: Estimating the Probability of an Event
probability number of Canadian license plates 21 total number of license plates 60
=
Try This: Example 1B
B. Josie sells TVs. On Monday she sold 13 plasma displays and 37 tube TVs. What is the probability that the first TV sold on Tuesday will be a plasma display? A tube TV?
P(plasma) = 0.26
P(plasma) + P(tube) = 1
0.26 + P(tube) = 1
P(tube) = 0.74
probability ≈ 1350
number of plasma displays total number of TVs
=13 + 37
13 =
Use the table to compare the probability that the Huskies will win their next game with the probability that the Knights will win their next game.
Additional Example 2: Application
Additional Example 2 Continued
The Knights are more likely to win their next game than the Huskies.
number of winstotal number of games
probability
probability for a Huskies win 13879 0.572
146probability for a Knights win 90 0.616
Use the table to compare the probability that the Huskies will win their next game with the probability that the Cougars will win their next game.
Try This: Example 2
Try This: Example 2 Continued
The Huskies are more likely to win their next game than the Cougars.
number of winstotal number of games
probability
probability for a Huskies win 13879 0.572
150probability for a Cougars win 85 0.567
Lesson Quiz: Part 1
1. Of 425, 234 seniors were enrolled in a math course. Estimate the probability that a randomly selected senior is enrolled in a math course.
2. Mason made a hit 34 out of his last 125 times at bat. Estimate the probability that he will make a hit his next time at bat.
0.27, or 27%
0.55, or 55%
Lesson Quiz: Part 2
3. Christina polled 176 students about their favorite ice cream flavor. 63 students’ favorite flavor is vanilla and 40 students’ favorite flavor is strawberry. Compare the probability of a student’s liking vanilla to a student’s liking strawberry.
about 36% to about 23%