Geometry HSAP Review, Mar 26

Probability

Book Sections: N/A

Essential Questions: What is simple probability, what are its components, where do I find it and how do I compute it? What is

multiple event probability?

Standards: N/A

Part 1

• Simple probability – One thing happening.

Notation

• The probability of an event will be

abbreviated as follows:

P(event) =

The Mathematical Definition of Probability

P(event) =

In words: The probability of an event is the ratio of favorable outcomes to the number of possible outcomes. That number will always be between 0 and 1.

Number of favorable outcomes

Total number of outcomes

What Are the Components of the Ratio

• Favorable outcomes – The number of ways within the sample space

that what you want to occur CAN occur

• Total number of outcomes – everything that can happen in the

sample space

The total number of outcomes is also known as all possible

outcomes

Some Simple Examples Total Number of Outcomes

If you roll a single fair die, there are 6 possible outcomes, which are

1, 2, 3, 4, 5, 6

Drawing a single card from a deck. There are 52 possible outcomes.

Flipping a coin has two possible outcomes, a head or a tail.

Some Simple Examples Favorable Outcomes

Betting on the number 5 on a roll of a die.

Selecting an even number on a roll of a die.

Selecting a red card from a deck of cards.

Selecting a queen from a deck of cards.

Selecting the 5 of diamonds from a deck of cards.

Calling heads on a coin flip.

Computing Probabilities

The number 5 on a roll of a die.

Selecting an even number on a roll of a die.

Selecting a red card from a deck of cards.

Selecting a queen from a deck of cards.

Selecting the 5 of diamonds from a deck of cards.

Calling heads on a coin flip.

More Examples

Expressing Probability

• Probability can be expressed as:

A fraction (in simplest form)

A decimal

A percent (%)

• Probability ranges between 0 and 1

Probability of 0 means the event is impossible

Probability of 1 means the event is a sure thing

Examples A bag contains 4 red, 3 blue, 2 green, and 1 yellow

marble. Compute the following probabilities based on

selecting a single marble from the bag:

Part 2

• Multiple event probability – More than one

thing happening.

The Probability of a Single Event

P(event) =

Number of favorable outcomes

Number of possible outcomes

Multiple Event Methodology

• Anytime we do multiple events, and they

involve the word and – meaning both, we

are multiplying probabilities.

Multiple Events

• Multiple Events – More than one random

event occurring simultaneously or in

succession.

• Multiple events are either independent or

dependent events

How Can I Tell the Difference

• Drawing or selecting anything from a pool

of objects and then selecting another

without replacing the first is an indication of

dependency

• A situation where there is a cause – effect

relationship

• Anything else, including selection with

replacement or from different sources, are

independent events

Independent or Dependent?

1. Tossing a coin and spinning a spinner

2. Drawing two cards from a single deck

3. Drawing two cards from separate decks

4. Selecting two marbles from a bag of marbles

5. Being a lifeguard and getting a suntan

6. Betting on different horses to win, place, and

show at the track

7. Rolling two dice

8. Parking in a no parking zone and getting a

parking ticket

I

D

I

D

D

D

I

D

Multiple Event Probabilities 1

• Multiple events are called event A and event B

A and B are independent

• P(A and B) = P(A) · P(B), P(A) and P(B) are simple

probabilities

An Example

P(7and 4)

P(even and odd)

P(yellow and red)

Multiple Event Probabilities 2

• Multiple events are called event A and event B

A and B are dependent

• P(A and B) = P(A) · P(B | A), where P(A) is a simple

probability and P(B | A) is a probability computed for B

given that A has already happened

Example • Bag A contains 10 marbles of the following colors:

4 green, 3 blue, 2 red, and 1 purple.

P(blue and red)

P(red and green)

P(green and gray)

P(green and green)

Adjusting a Sample Space

To compute P(B following A) – Two Steps to

adjusting a sample space:

1. There are now one less items in the space

2. The item ‘selected’ is not there, debit that

mini pool.

Multiple Event Probabilities

• Multiple events are called event A and event B

A and B are dependent

• P(A and B) = P(A) · P(B | A), where P(A) is a simple

probability and P(B | A) is a probability computed for B

given that A has already happened

A and B are independent

• P(A and B) = P(A) · P(B), P(A) and P(B) are simple

probabilities

Multiple - Multiple Events

• What if its more than two events?

• Assess the situation, compute probabilities,

keep multiplying

If dependent, think P(C | A and B)

Example

A person owns a collection of 30 CDs, of which 5 are country

music. If two CDs are selected randomly, what is the

probability that both are country music?

Working With All Multiple

Events

1. Determine dependency (Indep or Dep)

2. Visualize or draw a picture of sample space

3. Compute probability of first event

4. Compute probability of second event

Adjust sample space if a dependent event

5. Multiply probabilities together

6. Simplify (if required)

Homework: None

Class work: Handout CW 3/26, all