Probability

Basic Concepts of Probability

What you should learn:•How to identify the sample space of a probability experiment and to identify simple events.•How to distinguish among classical probability, empirical probability, and subjective probability.•How to identify and use properties of probability.

Probability Experiment-

Vocabulary

An experiment through which you obtain counts, measurement or responses.

Outcome - The result of a single trial

in a probability experiment.

Sample SpaceVocabulary

The set of all possible outcomes of a probability experiment.

Event-A subset of the sample space that consists of one or more outcomes of the probability experiment.

Identifying Parts of a Probability Experiment.Probability experiment

Sample Space

Event

Outcome

Roll a 6-sided die.

1,2,3,4,5,6

Rolling an even number

Rolling a 2

Simple Event-Vocabulary

An event that consists of a single outcome.

Example: Rolling a 2 on a die.

Non-Example: Selecting an Ace from a standard deck of cards….there are 4 aces.

Types of Probability

Classical (Theoretical) ProbabilityEmpirical (Statistical) Probability

Classical ProbabilityVocabulary

Used when each outcome in a sample space is equally likely to occur.

Probabilities can be expressed as fractions, decimals and percents.

FYI

This chapter, we will be expressing them as fractions or decimals rounded to three decimal places if necessary.

The probability of an impossible event is zero.

The probability of an event that is certain to occur is one. 0 P(E) 1 for any

event A.

Empirical ProbabilityVocabulary

Probability based on observations obtained from probability experiments.

Law of Large NumbersAs an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of that event.

Types of Probability

Classical (Theoretical) ProbabilityEmpirical (Statistical) Probability

Subjective Probability

Subjective ProbabilityVocabulary

Results from intuition, educated guesses and estimates.

Example: A company might predict that the chance of employees of the company going on strike is 25%.

There is no formula for Subjective Probability.

VocabularyComplement of an Event- The set of all outcomes in a

sample space that are not included in event E.

NotationE’- read as E prime

Properties of Probability

P(E) + P(E’) = 1

P(E) = 1 – P(E’)

P(E’) = 1 – P(E)

VocabularyOdds -

The ratio of the number of successful outcomes to the number of unsuccessful outcomes.

Assignment: pg. 111: 1-26,29

Conditional Probability and the Multiplication Rule

What you should learn:•How to find the probability of an event given that another event has occurred.•How to distinguish between independent and dependent events.•How to use the multiplication rule to find the probability of two events occurring in sequence.•How to use the multiplication rule to find conditional probabilities.

Conditional ProbabilityVocabulary

The probability of an event occurring, given that another event has already occurred.

Notation

Example: Two cards are selected in a sequence from a standard deck of 52 cards. Find the probability that the second card is a queen, given that the first card is a king and wasn’t replaced before the second drawing occurred.

Examples of Conditional Probability

Types of Conditional Probability

Independent Events

Dependent Events

Independent EventsVocabulary

If the occurrence of one of the events does not affect the probability of the occurrence of the other events.

Dependent EventsIf the occurrence of one of the events affects the probability of the occurrence of the other events.

The Multiplication Rule

VocabularyThe probability that 2 events A and B will occur in sequence is P(A and B) = P(A) ∙ P(B|A)

Assignment: pg. 119: 1,5-10,13-

16,18-20

The Addition Rule

What you should learn:•How to determine if two events are mutually exclusive.•How to use the addition rule to find the probability of two events.

Mutually ExclusiveVocabulary

Two events are mutually exclusive if they can not occur at the same time.

A B

A and B are mutually exclusive.

A B

A and B are not mutually exclusive.

Eligible voters and 10 year olds.

Examples of Mutually Exclusive Events

voters 10 yr. olds

No overlap

Jacks in a deck and threes in a deck

Examples of Mutually Exclusive Events

jacks threes

No overlap

Spinning a spinner and rolling a dice.

Examples of Mutually Exclusive Events

spins numbers

No overlap

Jacks and diamonds

Examples of Non-Mutually Exclusive Events

jacks diamonds

overlap

j

Sophomores and boys

Examples of Non-Mutually Exclusive Events

10th graders boys

overlap

10th gr. boys

The Addition RuleThe probability that events A or B will occur is given by…

Where P(A and B) would be the overlap section.

Assignment: pg. 129: 2-18

Counting Principles

What you should learn:•How to use the Fundamental Counting Principle to find the number of ways two or more events can occur.•How to find the number of ways a group of objects can be arranged in order.•How to find the number of ways to choose several objects from a group without regard to order.•How to use counting principles to find probabilities.

The Fundamental Counting Principle -

VocabularyIf one event can occur in m ways, and a second event can occur in n ways, the number of ways the 2 events can occur in sequence is m∙n.

This rule can be extended for any number of events occurring in sequence.

VocabularyFactorial -

A multiplication pattern denoted by n!. It is the product of n with each of the positive counting numbers less than n.

n! = n(n-1)(n-2)(n-3)…(1)

Special Definition: 0! = 1

Permutation -Vocabulary

An ordered arrangement of objects.

The number of permutations of n distinct objects is n!.

Example of a PermutationHow many permutations are their for 3 people to fill 3 vacant positions at Corporation Z?There would be 3! arrangements to fill these vacancies.3! = 3∙2∙1 = 6 arrangements

A,B,C

A,C,B

B,A,C

B,C,A

C,A,B

C,B,A

Permutation of n items taken r at a time-

What if I don’t want to use all the items.

Keep in mind….Order is Important.

Permutation of n items taken r at a time with duplicates-

Distinguishable Permutations

Keep in mind….Order is still Important.

Combination -Vocabulary

An arrangement of objects where order does not matter.

Suppose you want to buy 3 CDs from a selection of 5?

Suppose you want to buy 3 CDs from a selection of 5?

Example of a Combination

The tree diagram would lead to the following outcomes.

ABCABDABE

ACBACDACE

ADBADCADE

AEBAECAED

BACBADBAE

BCABCDBCE

BDABDCBDE

BEABECBED

CABCADCAE

CBACBDCBE

CDACDBCDE

CEACEBCED

DABDACDAE

DBADBCDBE

DCADCBDCE

DEADEBDEC

EABEACEAD

EBAEBCEBD

ECAECBECD

EDAEDBEDC

Now take care of

duplicates…There are 10 combinations

.

There has to be an easier way!!!

Combinations

Assignment: pg. 140: 1-28

Application of the Counting Principles

You can determine probabilities if your can determine how many ways a particular event can occur.

Assignment: pg. 142: 29-30