Introductory Statistics

Lesson 3.1 A

Objective: SSBAT identify sample space and find probability of simple events.

Standards: M11.E.3.1.1

Probability Measures how likely it is for something to occur

A number between 0 and 1

Can be written as a fraction, decimal or percent

Probability equal to 0 Impossible to happenProbability equal to 1 Will definitely occur

Probability is used all around us and can be used to help make decisions.

Weather“There is a 90% chance it will rain tomorrow.”

You can use this to decide whether to plan a trip to the amusement park tomorrow or not.

Surgeons“There is a 35% chance for a successful surgery.”

They use this to decide if you should proceed with the surgery.

Probability Experiment

An action, or trial, through which specific results (counts, measurements, or responses) are obtained.

Outcome

The result of a single trial in an experiment

Example: Rolling a 2 on a die

Sample Space

The set of ALL possible outcomes of a probability experiment.

Example: Experiment Rolling a Die

Sample Space: 1, 2, 3, 4, 5, 6

Event A subset (part) of the sample space. It consists of 1 or more outcomes

Represented by capital letters

Example: Experiment Rolling a Die

Event A: Rolling an Even Number

Tree Diagram

A method to list all possible outcomes

Examples: Find each for all of the followinga) Identify the Sample Spaceb) Determine the number of outcomes

1. A probability experiment that consists of Tossing a Coin and Rolling a six-sided die.

a) Make a tree diagram

Examples: Find each for all of the followinga) Identify the Sample Spaceb) Determine the number of outcomes

1. A probability experiment that consists of Tossing a Coin and Rolling a six-sided die.

a) Make a tree diagram

H T

1 2 3 4 5 6 1 2 3 4 5 6

Sample Space: {H1, H2, H3, H4, H5, H6,T1, T2, T3, T4, T5, T6}

Examples: Find each for all of the followinga) Identify the Sample Spaceb) Determine the number of outcomes

1. A probability experiment that consists of Tossing a Coin and Rolling a six-sided die.

b) There are 12 outcomes

2. An experimental probability that consists of a person’s response to the question below and that person’s gender.

Survey Question: There should be a limit on the number of terms a U.S. senator can serve.Response Choices: Agree, Disagree, No Opinion

a)

Sample Space: {FA, FD, F NO, MA, MD, M NO}

b) There are 6 outcomes

3. A probability experiment that consists of tossing a coin 3 times.

a)

{HHH, HHT, HTH, HTT, THH, THT, TTH, TTH, TTT}

b) There are 8 outcomes

Fundamental Counting Principle

A way to find the total number of outcomes there are

It does not list all of the possible outcomes – it just tells you how many there are

If one event can occur in m ways and a second event can occur n ways, the total number of ways the two events can occur in sequence is m·n

This can be extended for any number of events

In other words:

The number of ways that events can occur in sequence is found by multiplying the number of ways each event can occur by each other.

Take a look at a previous example and solve using the Fundamental Counting Principle.

How many outcomes are there for Tossing a Coin and Rolling a six sided die?

There are 2 outcomes for the coinThere are 6 outcomes for the die

Multiply 2 times 6 together to get the total number of outcomes

Therefore there are 12 total outcomes.

1. You are purchasing a new car. The possible manufacturers, car sizes, and colors are listed below. How many different ways can you select one manufacturer, one car size, and one color?

Manufacturer: Ford, GM, HondaCar Size: Compact, MidsizeColor: White, Red, Black, Green

3 · 2 · 4 = 24

There are 24 possible combinations.

2. The access code for a car’s security system consists of four digits. Each digit can be 0 through 9 and the numbers can be repeated.

there are 10 possibilities for each digit

10 · 10 · 10 · 10 = 10,000

There are 10,000 possible access codes.

3. The access code for a car’s security system consists of four digits. Each digit can be 0 through 9 and the numbers cannot be repeated.

There are 10 possibilities for the 1st number and then subtract 1 for the next amount and so on

10 · 9 · 8 · 7 = 5040

There are 5,040 possible access codes.

4. How many 5 digit license plates can you make if the first three digits are letters (which can be repeated) and the last 2 digits are numbers from 0 to 9, which can be repeated?

there are 26 possible letters and 10 possible numbers

26 · 26 · 26 · 10 · 10 = 1,757,600

There are 1,757,600 possible license plates

5. How many 5 digit license plates can you make if the first three digits are letters, which cannot be repeated, and the last 2 digits are numbers from 0 to 9, which cannot be repeated?

26 · 25 · 24 · 10 · 9 = 1,404,000

There are 1,404,000 possible license plates

6. How many ways can 5 pictures be lined up on a wall?

5 · 4 · 3 · 2 · 1

There are 120 different ways.

Simple Event

An event that consists of a single outcome

Example of a Simple Event

Rolling a 5 on a die - There is only 1 outcome, {5}

Example of a Non Simple Event

Rolling an Odd number on a die – There are 3 possible outcomes: {1, 3, 5}

Determine the number of outcomes in each event. Then decide whether each event is simple or not?

1. Experiment: Rolling a 6 sided die Event: Rolling a number that is at least a 4

There are 3 outcomes (4, 5, or 6)

Therefore it is not a simple event

Determine the number of outcomes in each event. Then decide whether each event is simple or not?

2. Experiment: Rolling 2 dice Event: Getting a sum of two

There is 1 outcome (getting a 1 on each die)

Therefore it is a simple event

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