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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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Basics of Probability
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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Basics of Probability
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
S = {1, 2, 3, 4, 5, 6} nS = 6
E= {2, 4, 6} nE = 3
3n EP E
n S
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Basics of Probability
Example: A single die is rolled. Write each event in set
notation and give the probability of the event.
(b) the number showing is greater than 4
E= {5, 6} nE = 2
! !2 1
6 3P E
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Basics of Probability
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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Basics of Probability
Example: Find the probability ofnotdrawing an ace
from a well-shuffled deck of cards.
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Basics of Probability
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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Binomial Probability
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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Binomial Probability
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