1 Chapter 9 Introducing Probability
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Chapter 9
Introducing Probability
From Exploration to Inferencep. 150 in text
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The Idea of Probability• Probability helps
us deal with “chance”
• Definition: the probability of an event is its expected proportion in an infinite infinite series of repetitions
Example: A random sample of n = 100 children has 8 individuals with asthma. What is the probability a child has asthma?
ANS: We do not know. Although 8% is a reasonable “guesstimate,” the true probability is not known because our sample was not infinitely large
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How Probability BehavesCoin Toss Example
The proportion of heads approaches
0.5 with many, many tosses.
Chance behavior is unpredictable in the short run, but is predictable in the long run.
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Probability models consist of these two parts:1)1) Sample SpaceSample Space (S) = the set of all possible
outcomes of a random process2)2) ProbabilitiesProbabilities (Pr) for each possible outcome in
the sample space
Probability Models
Example of a probability model
“Toss a fair coin once”
S = {Heads, Tails} all possible outcomes
Pr(heads) = 0.5 and Pr(tails) = 0.5 probabilities for each outcome
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4 Basic Rules of Probability (Summary)
1.1. 0 0 ≤ Pr(A) ≤ 1≤ Pr(A) ≤ 1
2.2. Pr(S) = 1Pr(S) = 1
3.3. Addition Rule for Disjoint EventsAddition Rule for Disjoint Events
4.4. Law of ComplementsLaw of Complements
Also on bottom of page 1 of Formula Sheet
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Rule 1 (Range of Possible Probabilities)
Let A ≡ event APr(A) ≡ probability of event A
Rule 1 says Rule 1 says ““0 0 ≤ Pr(A) ≤ 1≤ Pr(A) ≤ 1”” Probabilities are Probabilities are alwaysalways between 0 & 1 between 0 & 1
Pr(A) = 0 means A never occurs
Pr(A) = 1 means A always occurs
Pr(A) = .25 means A occurs 25% of the time
Pr(A) = 1.25 Impossible! Must be something wrong
Pr(A) = some negative number Impossible! Must be something wrong
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Rule 2 (Sample Space Rule)Let S ≡ the Sample Space
Pr(S) = 1Pr(S) = 1
All probabilities in the sample space All probabilities in the sample space must sum to 1 exactly.must sum to 1 exactly.
Example: “toss a fair coin”
S = {heads or tails}
Pr(heads) + Pr(tails) = 0.5 + 0.5 = 1.0
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Events A and B are disjoint if they can Events A and B are disjoint if they can never occur together. never occur together.
When events are disjoint:When events are disjoint:Pr(A or B) = Pr(A) + Pr(B)Pr(A or B) = Pr(A) + Pr(B)
Age of mother at first birthLet A ≡ first birth at age < 20: Pr(A) = 25%Let B ≡ first birth at age 20 to 24: Pr(B) = 33%Let C ≡ age at first birth ≥25 Pr(C) = 42%
Probability age at first birth ≥ 20 = Pr(B or C) = Pr(B) + Pr(C) = 33% + 42% = 75%
Rule 3 (Addition Rule, Disjoint Events)
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Rule 4 (Rule of Complements)Let Ā ≡ A does NOT occur
This is called the complementcomplement of event A
Pr(Ā) = 1 – Pr(A)Pr(Ā) = 1 – Pr(A)
Example:
If A ≡ “survived” then Ā ≡ “did not survive”
If Pr(A) = 0.9 then Pr(Ā) = 1 – Pr(A) = 1 – 0.9 = 0.1
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Probability Mass Functions pmfs
Example of a pmf: A couple wants three children. Let X ≡ the number of girls they will haveHere is the pmf that suits this situation:
Probability mass functions are made up of a list of separated outcomes.
For discrete random variables.
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Probability Density Functions pdfs (“Density Curves”)
• To assign probabilities for continuous random variables we density curvedensity curve
• Properties of a density curve– Always on or above horizontal axis
– Has total area under curvearea under curve (AUCAUC) of exactly 1
– AUCAUC in any range = probability of a value in that range
Probability density functions form a continuum of possible outcomes.For continuous random variables.
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Example of a pdf
Note• The curve is always on or above horizontal axis and has AUC =
height × base = 1 × 1 = 1• Probability = AUC in the range. Examples follow.• Pr(X < .5) = height × base = 1 × .5 = .5• Pr(X > 0.8) = height × base = 1 × .2 = .2 • Pr (X < .5 or X > 0.8) = .5 + .2 = .7
This random spinner has this pdf density “curve”
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pdf Density Curves• Density curves come in many shapes
– Prior slide showed a “uniform” shape– Below are “Normal” and “skewed right” shapes
• Measures of center apply to density curves– µ (expected value or “mean”) is the center balancing point– Median splits the AUC in half
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From Histogram to Density Curve
• Histograms show distribution in chunks
• The smooth curve drawn over the histogram represents a Normal density curve for the distribution
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Area Under the Curve (AUC)
30% of students had scores ≤ 6
30% 70%
Area in Bars = proportion in that range30% of students
had scores ≤ 6
Shaded area = 30% of total area of the histogram
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Area Under the Curve (AUC)
30%
Area Under Curve = proportion in that range!
30% of students had scores ≤ 6
30% of area under the curve (AUC) is shaded
70%
Summary of Selected Points
• To date we have studied descriptive statisticsdescriptive statistics. From here forward we study inferential statisticsinferential statistics {2}
• ProbabilityProbability is the study chance; chance is unpredictable in the short run but is predictable in the long run {3 - 4}; take the rules of probability to heart{5 - 10}
• Discrete random variablesDiscrete random variables are described with probability mass function
• Continuous random variablesContinuous random variables are described with density curves density curves with the area under the curve curve (AUC)(AUC) corresponding to probabilities