Transcript Probabilities

Chapter 9
Introducing
Probability
1
From Exploration to Inference
p. 150 in text
Ch 1 – Ch 5:
Ch 7+:
Purpose: Unrestricted
exploration & searching
for patterns
Conclusions are informal
and apply only to current
data
Purpose:
Answer specific
questions
Conclusions are formal
and apply to broad class
of circumstances
The Idea of Probability
• Probability
helps us deal
with “chance”
• Definition: the
probability of an
event is its
expected
proportion in an
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
3
How Probability Behaves
Coin Toss Example
Chance
behavior is
unpredictable
in the short
run, but is
predictable in
the long run.
The proportion of
heads approaches
0.5 with many,
many tosses.
4
Probability Models
Probability models consist of these two parts:
1) Sample Space (S) = the set of all possible
outcomes of a random process
2) Probabilities (Pr) for each possible outcome in
the sample space
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.
2.
3.
4.
0 ≤ Pr(A) ≤ 1
Pr(S) = 1
Addition Rule for Disjoint Events
Law of Complements
Also on bottom of page 1 of Formula Sheet
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Rule 1 (Range of Possible Probabilities)
Let A ≡ event A
Pr(A) ≡ probability of event A
Rule 1 says “0 ≤ Pr(A) ≤ 1”
Probabilities are always 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) = 1
All probabilities in the sample space
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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Rule 3 (Addition Rule, Disjoint Events)
Events A and B are disjoint if they can
never occur together.
When events are disjoint:
Pr(A or B) = Pr(A) + Pr(B)
Age of mother at first birth
Let 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%
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Rule 4 (Rule of Complements)
Let Ā ≡ A does NOT occur
This is called the complement of event 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
Probability mass functions are made up of a
list of separated outcomes.
For discrete random variables.
Example of a pmf:
A couple wants three children.
Let X ≡ the number of girls they will have
Here is the pmf that suits this situation:
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Probability Density Functions pdfs
(“Density Curves”)
Probability density functions form a continuum of
possible outcomes.
For continuous random variables.
• To assign probabilities for continuous random
variables  we density curve
• Properties of a density curve
– Always on or above horizontal axis
– Has total area under curve (AUC) of exactly 1
– AUC in any range = probability of a value in that
range
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Example of a pdf
This random spinner has this pdf density “curve”
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
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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
14
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
3/28/2016
15
Area Under the Curve (AUC)
Area in Bars =
proportion in that
range
30% of students
had scores ≤ 6
Shaded area =
30% of total area
of the histogram
30%
3/28/2016
70%
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Area Under the Curve (AUC)
Area Under
Curve =
30% of students
had scores ≤ 6
proportion in that
range!
30% of area
under the curve
(AUC) is shaded
30%
3/28/2016
70%
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Summary of Selected Points
• To date we have studied descriptive statistics.
From here forward we study inferential statistics
{2}
• Probability 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 variables are described with
probability mass function
• Continuous random variables are described with
density curves with the area under the curve
(AUC) corresponding to probabilities