Goals
Notes
Chapter 3: Concepts of probabiliy.
Topics:
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Basic Probability and Counting
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Conditional Probability and Multiplication Rules
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Mutual Exclusivity and Additional Rules
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Permutations and Combinations
Section 3.1: Basic Probability and Counting
Goals:
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Probability Experiments
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Fundamental Counting
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Complentary Events
Probability Experiments
Probability experiments consist of an action with specific results
(counts, responses, or measurements), obtained by random
phenomenon. All possible outcomes are listed as a set called a
sample space. Subsets of the sample space are called events.
Notes
Example 1:
Probability experiment: ”Flip a coin two times.”
Sample space: {HH, HT , TH, TT }
Event: ”Get two heads, {HH}.”
Outcome: Flipped TH, {TH}.
The event {HH} is a simple event because it has exactly one possible
outcome.
Example 2:
Probability experiment: ”Roll a six-sided die.”
Sample space: {1, 2, 3, 4, 5, 6}.
Event: ”Roll an even number, {2, 4, 6}”
Outcome: Roll a 5, {5}.
The event {2, 4, 6} is a not simple event because it has more than
one possible outcome.
Example 3:
Probability experiment: ”Roll a six-sided die.”
Sample space: {1, 2, 3, 4, 5, 6}.
Event: ”Roll at least a 5, {5, 6}”
Outcome: Roll a 5, {5}.
The event {5, 6} is a not simple event because it has more than one
possible outcome.
Notes
Fundamental Counting Principle
When one event can occur m ways and another event can occur n
ways, the number of ways the two events can occur in sequence is
m · n (multiplication).
Notes
Example 1: A restaurant offers three main dishes and four desserts.
A meal comes with one main dish and one dessert. How many
different ways can you select a meal?
Example 2: A license plate with three characters is made from
selecting a letter, then selecting two numerical digits. There are 26
letters and 10 digits. How many different license plates can be made?
Tip: Drawing a diagram to show choices can help.
I flip a coin that comes up heads by 50%. If I flip it 100 times, should
I get 50 heads and 50 tails?
Notes
The Law of Large Numbers
As an experiment is repeated over and over, the empirical probability
of an event approaches the theoretical probability of the event.
In a simulation, I get 58 heads, obtaining an emprical probabilty of
58
. Tossing the coin a large number of times will give me a
100
proportion very close to 12 .
n tosses
10
50
100
500
1 Million
Experiment 1
5
13
58
269
501348
Experiment 2 Experiment 3
5
6
21
19
47
49
274
270
4999557
500697
Complements
Let E be an event. The complement of an event E 0 is the set of all
outcomes in a sample space that are not included in event E .
Because 0 ≤ P(E ) ≤ 1,
Total Probability
I P(E ) + P(E 0 ) = 1
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P(E ) = 1 − P(E 0 )
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P(E 0 ) = 1 − P(E )
Example A standard deck of cards has 52 cards. There are four suits
(♣, ♥, ♦, ♠). One card is drawn from the deck. What is the
complement of the event E : ”Selecting a heart” ? What is the
probability of E , and the probability of E 0 ?
Notes
Assignment
Notes
For June 16 - 2014
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Suggested: p 138 #17, #20, #23, #37, #51, #63
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Required: p 138 #1, #4, #5, #6, #46, #65, #75, #76, #77
(Due: June 18)
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Read 3.1 and 3.2.
Notes
Notes