统计学 第三章

H
Sample Space Examples
Experiment
• • • • • • • Toss a Coin, Note Face Toss 2 Coins, Note Faces Select 1 Card, Note Kind Select 1 Card, Note Color Play a Football Game Inspect a Part, Note Quality Observe Gender
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Probabilities
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What is Probability?
1. Numerical measure of the likelihood that event will cccur • P(Event) • P(A) • Prob(A) 2. Lies between 0 & 1 3. Sum of sample points is 1 0
Events
1. Specific collection of sample points 2. Simple Event • Contains only one sample point 3. Compound Event • Contains two or more sample points
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Thinking Challenge
• What’s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing). • So toss a coin twice. Do it! Did you get one head & one tail? What’s it all mean?
Compound Events
Compound events: Composition of two or more other events. Can be formed in two different ways.
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Unions & Intersections
3.1
Events, Sample Spaces, and Probability
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Experiments & Sample Spaces
1. Experiment
• Process of observation that leads to a single outcome that cannot be predicted with certainty
HH TT
HT
S
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Event Examples
Experiment: Toss 2 Coins. Note Faces. Sample Space: HH, HT, TH, TT Event • • • • 1 Head & 1 Tail Head on 1st Coin At Least 1 Head Heads on Both Outcomes in Event HT, TH HH, HT HH, HT, TH HH
© 1984-1994 T/Maker Co.
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Steps for Calculating Probability
1. Define the experiment; describe the process used to make an observation and the type of observation that will be recorded 2. List the sample points 3. Assign probabilities to the sample points 4. Determine the collection of sample points contained in the event of interest 5. Sum the sample points probabilities to get the event probability
Ace
Black
S
Event Ace ∪ Black: A♥, ..., A♠, 2♣, ..., K♠
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Event Ace: A ♥, A ♦, A ♣, A ♠
Event Union: Two–Way Table
Experiment: Draw 1 Card. Note Kind, Color & Suit. Color Simple
Impossible
1
Certain
.5
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Probability Rules for Sample Points
Let pi represent the probability of sample point i. 1. All sample point probabilities must lie between 0 and 1 (i.e., 0 ≤ pi ≤ 1). 2. The probabilities of all sample points within a sample space must sum to 1 (i.e., Σ pi = 1).
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Equally Likely Probability
P(Event) = X / T
• X = Number of outcomes in the event • T = Total number of sample points in Sample Space • Each of T sample points is equally likely — P(sample point) = 1/T
2. Sample point
• Most basic outcome of an experiment
Sample Space Depends on Experimenter!
3. Sample space (S)
• Collection of all possible outcomes
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n! = n (n − 1)(n − 2 )L (3)(2 )(1)
For example, 5! = 5 ⋅ 4 ⋅ 3⋅ 2 ⋅1
0! is defined to be 1.
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3.2
Unions and Intersections
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Many Repetitions!*
Total Heads Number of Tosses
1.00 0.75 0.50 0.25 0.00 0 25 50 75 100 125
Number of Tosses
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© 2011 Pearson Education, Inc
Venn Diagram
Experiment: Toss 2 Coins. Note Faces.
Sample Space S = {HH, HT, TH, TT}
Compound Event: At least one Tail
TH
Outcome
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© 1984-1994 T/Maker Co.
Experiment: Observe Gender
Visualizing Sample Space
1. Listing
S = {Head, Tail}
2.
Venn Diagram
T S
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Sample Space Properties
1. Mutually Exclusive • 2 outcomes can not occur at the same time — Male & Female in same person 2. Collectively Exhaustive • One outcome in sample space must occur. — Male or Female
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Learning Objectives
1. Develop probability as a measure of uncertainty 2. Introduce basic rules for finding probabilities 3. Use probability as a measure of reliability for an inference
1. Union
• • • • • • Outcomes in either events A or B or both ‘OR’ statement Denoted by ∪ symbol (i.e., A ∪ B) Outcomes in both events A and B ‘AND’ statement Denoted by ∩ symbol (i.e., A ∩ B)
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Combinations Rule
A sample of n elements is to be drawn from a set of N elements. The, the number of different samples possible ⎛ N⎞ is denoted by ⎜ ⎟ and is equal to ⎝n ⎠ ⎛ N⎞ N! ⎜ n ⎟ = n!(N − n )! ⎝ ⎠ where the factorial symbol (!) means that
Statistics for Business and Economics
Chapter 3 Probability
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Contents
1. 2. 3. 4. Events, Sample Spaces, and Probability Unions and Intersections Complementary Events The Additive Rule and Mutually Exclusive Events 5. Conditional Probability 6. The Multiplicative Rule and Independent Events
Sample Space
{Head, Tail} {HH, HT, TH, TT} {2♥, 2♠, ..., A♦} (52) {Red, Black} {Win, Lose, Tie} {Defective, Good} {Male, Female}
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© 2011 Pearson Education, Inc
2. Intersection
Event Union: Venn Diagram
Experiment: Draw 1 Card. Note Kind, Color & Suit.
Sample Space: 2♥, 2♦, 2♣, ..., A♠ Event Black: 2♣, 2♠, ..., A♠