Math-2

Lesson 10-1 Lesson 10-1 ProbabilityProbability

Definitions

Sample Space: the set of all possible outcomes for an experiment.

Outcome: A possible result of a probability experiment is called an outcome

You may have noticed that for each of the experiments above, the sum of the probabilities of each outcome is 1. This is no coincidence.

The sum of the probabilities of the distinct outcomes within a sample space is 1.

Diagrams• Diagrams are often the key to getting started

on a problem. They can clarify relationships that appear complicated when written.

• A tree diagram is a type of systematic list that is used to organize information spatially.

___ ___ ______ ___ ___

Given the first letter above, the second letter could be either:Given the first letter above, the second letter could be either:

Arranging 3 Objects in OrderHow many ways can you arrange the lettersHow many ways can you arrange the letters AA, , BB, and , and CC in order? in order?

AA

BB or or CC

Any one of the 3 letters could be the Any one of the 3 letters could be the 11stst letter. letter.

The only option for the 3The only option for the 3rdrd letter in this case is: letter in this case is: CC

AABBCC

BB CC

Given the first letter above, the second letter could be:Given the first letter above, the second letter could be:

Arranging 3 Objects in OrderHow many ways can you arrange the lettersHow many ways can you arrange the letters AA, , BB, and , and CC ? ?

AA BB CC

BB or or CC AA or or CC AA or or BB

Any one of the following 3 could be the Any one of the following 3 could be the 11stst letter. letter.

The only option for the 3The only option for the 3rdrd letter in each case is: letter in each case is:

AABBCC AACCBB BBAACC BBCCAA CCBBAA CCAABB

SIX waysSIX ways

“Tree diagram” How many way can you How many way can you arrange the lettersarrange the letters AA, , BB, and , and CC ? ?

AA BB CC

BB CC AA CC BB AA

CC BB CC AA AA BB

AABBCC,,AACCBB,,

BBAACC,,

BBCCAA,,CCBBAA,,

CCAABBSIX waysSIX ways

We call this a We call this a tree diagramtree diagram..

AABBCC

Multiplication PrincipleMultiplication Principle states: If an event occurs in m ways and

another event occurs independently in n ways, then the two events can occur in m × n ways.

What is the sample space?

Sample Space: All of the experimental outcomes

Your Turn

1. Draw a tree diagram representing the outcomes for flipping a coin then tossing a die.

• a. How many outcomes are there?• b. What is the probability of getting heads, then

an even number?

• *this is problem 1 on your homework

Using the Multiplication PrincipleIf a license plate has three letters followed by three If a license plate has three letters followed by three

numerical digits. Find the number of different numerical digits. Find the number of different license plates that could be formed if there is no license plates that could be formed if there is no restriction on the letters or digits that can be used.restriction on the letters or digits that can be used.

L L L # # # L L L # # #

How many possibilities for the 1How many possibilities for the 1stst position (letter)? position (letter)?

26 26

Using the Multiplication Principle

L L L # # # L L L # # #

How many possibilities for the 2How many possibilities for the 2ndnd position? position?

26 26 * 26 * 26

Using the Multiplication Principle

L L L # # # L L L # # #

How many possibilities for the 3How many possibilities for the 3rdrd position? position?

26 26 * 26 * 26 * 26 * 26

Using the Multiplication Principle

L L L # # # L L L # # #

How many possibilities for the 4How many possibilities for the 4 thth position (number)? position (number)?

26 26 * 26 * 26 * 26 * 26 * 10 * 10

Using the Multiplication Principle

L L L # # # L L L # # #

How many possibilities for the 5How many possibilities for the 5 thth position? position?

26 26 * 26 * 26 * 26 * 26 * 10 * 10 * 10 * 10

Using the Multiplication Principle

L L L # # # L L L # # #

How many possibilities for the 6How many possibilities for the 6 thth position? position?

26 26 * 26 * 26 * 26 * 26 * 10 * 10 * 10 * 10 * 10 * 10

Total number of distinct license plates = Total number of distinct license plates = 17,576,000 17,576,000

Counting the number of ways to arrange songs on a CD:

___ ___ ___ ___ ___ ______ ___ ___ ___ ___ ___991010 88

How many options are there for the 1How many options are there for the 1stst song? song?

10 songs fit into 6 spots on the demo disk.10 songs fit into 6 spots on the demo disk.

How many options are there for the 2How many options are there for the 2ndnd song? song?

33rdrd song? song? 44thth song?song? 55thth song song?? 66thth song? song?

5577 66

The “multiplication principle.”The “multiplication principle.”When arranging things in order (letters A, B, and C), the When arranging things in order (letters A, B, and C), the totaltotal number of possible ways to arrange things is the number of possible ways to arrange things is the productproduct of the number of possibilities for each step.of the number of possibilities for each step.

Probability“What’s the chance of something

happening?”

““There is a 100% chance it will rain today.”There is a 100% chance it will rain today.”

““There is less than a 5% chance you will be picked.There is less than a 5% chance you will be picked.

Think about “Numerical Measure”

Can probability be equal to 50%?Can probability be equal to 50%?

What is the largest number that a probability can be?What is the largest number that a probability can be?

What is the smallest number that a probability can be?What is the smallest number that a probability can be?

Can there be a (– 20)% chance something will happen?Can there be a (– 20)% chance something will happen?

ProbabilityWhen When discussingdiscussing probability, you can use either probability, you can use either “%”, fraction, or the decimal equivalent. “%”, fraction, or the decimal equivalent.

“ “There is a 40% chance of thunderstorms today.”There is a 40% chance of thunderstorms today.”

In mathematicsIn mathematics, we convert % to the decimal , we convert % to the decimal equivalent or leave it in fraction form.equivalent or leave it in fraction form.

“ “The probability of rain today is 0.4.”The probability of rain today is 0.4.”

Your Turn 2. The probability that a student passes the drivers ed.

Written test is 62%. The probability that a student passes the driving part of the test is 86%. Draw a tree diagram showing the different outcomes (two branches then two more branches)

• a. Find the end probability for each of the branches of your diagram.

• b. Add up all the end probabilities• c. What is the probability that the student passes both tests?• d. What is the probability that a student passes only one of the

tests?• *this is problem 2 on your homework

Again

• 3. Every day that you drive to school you break the speed limit. The probability that you won’t get a ticket is 98%. Wahoo! You think that you will never get a ticket! You will make 180 trips to school this year.a. What is the probability that you won’t get any

tickets for the whole year? (use percents)b. What is the probability that you will get at least

one ticket? (use percents)*this is problem 3 on your homework

Vocabulary Review

• Numerical Measure• Experiments• Sample Space• Outcomes• Diagrams• Tree Diagrams• Multiplication Principal• Factorials

Homework 10.1

• HW 10.1 Basic Probability • Story Problems 1-7

– *use tree diagrams to solve problems.