9-2 Basics of Probability (Presentation)

8/6/2019 9-2 Basics of Probability (Presentation)

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9-2 Basics of Probability

Unit 9 Probability & Mathematical Induction


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Concepts and Objectives

Basics of Probability (Obj. #33)

Calculate the probability of an event

Use the complement to calculate probability

Calculate the probability of two or more events

Calculate the binomial probability of an event


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Basics of Probability

The setSof all possible outcomes of a given experiment

is called the sample space of the experiment.

Any subset of the sample space is called an event.

In a sample space with equally likely outcomes,

theprobabilityof an eventE, written PE, is theratio of the number of outcomes in sample space S

that belong to eventE, nE, to the total number ofoutcomes in sample space S, nS. That is,

!n E

P En S


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Example: A single die is rolled. Write each event in set

notation and give the probability of the event.

(a) the number showing is even

(b) the number showing is greater than 4


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(a) the number showing is even

S = {1, 2, 3, 4, 5, 6} nS = 6

E= {2, 4, 6} nE = 3

3n EP E

n S


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(b) the number showing is greater than 4

E= {5, 6} nE = 2

! !2 1

6 3P E


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If an event is certain to occur, then the probability will

be 1. If it is impossible for an event to occur, then the

probability is 0.

Therefore, for any eventE, PE will always be between 0and 1 inclusive.

The set of all outcomes in the sample space that do not

belong to eventEis called the complementofE, written

E. The probability ofE is 1 PE.


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Example: Find the probability ofnotdrawing an ace

from a well-shuffled deck of cards.


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Example: Find the probability ofnotdrawing an ace

from a well-shuffled deck of cards.

! ! !4 1drawing an ace52 13

P E P

' 1P E P E

!1

113 !

12

13


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Union of Two (or More) Events

Since events are sets, we can use set operations to find

the union of two events.

Suppose a fair die is rolled. LetHbe the event the result

is a 2, and Kthe event the result is an even number.

H= {2} K= {2, 4, 6} H K= {2, 4, 6}

Notice that

1

6P H

1

2

P K

1

2

P H K

{ P H P K P H K


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Union of Two or More Events

From our previous problem:

For any events Eand F,

! ! orP E F P E F P E P F P E F

P H K P H P K P H K

1 1 1

6 2 6!

1

2


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Union of Two or More Events

Example: Suppose two fair dice are rolled. Find the

probability that the first die shows a 2, or the sum of the

two dice is 6 or 7.

A = {2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6}

B = {1, 5, 1, 6, 2, 4, 2, 5, 3, 3, 3, 4, 4, 2, 4, 3,5, 1, 5, 2, 6, 1}

! !6 136 6P A ! 1136

P B ! !2 136 18P A B

! 1 11 16 36 18

P A B !5

12


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Binomial Probability

Abinomial experimentis an experiment that consists of

repeated independent trials with only two outcomes in

each trial, success or failure. Let the probability of

success in one trial bep. Then the probability of failure

is 1 p, and the probability of exactly rsuccesses in n

trials is given by

An easier calculator method is and enter n,p, and r.

1n rr

np p

r


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Example: An experiment consists of rolling a die 10

times. Find the probability that in exactly 4 of the rolls,

the result is a 3.


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Example: An experiment consists of rolling a die 10

times. Find the probability that in exactly 4 of the rolls,

the result is a 3.

P= .054266

! ! !110, ,6

n p r


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Homework

College Algebra (brown book)

Page 1061: 9-24 (v3s), 33

Turn in: 15, 18, 24, 35

Classwork: Algebra & Trigonometry(green book)

Page 666: 2-6

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9-2 Basics of Probability (Presentation)