Transcript 10: Introducing probability

Chapter 10
Introducing Probability
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Idea of Probability
• Probability is the
science of chance
behavior
• Chance behavior is
unpredictable in the
short run, but is
predictable in the long
run
• The probability of an
event is its expected
proportion in an
infinite series of
repetitions
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The probability of any
outcome of a random
variable is an expected
(not observed) proportion
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How Probability Behaves
Coin Toss Example
Eventually, the
proportion of
heads
approaches 0.5
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How Probability Behaves
“Random number table example”
The probability of a “0” in Table B is 1 in 10
(.10)
Q: What proportion of the first 50 digits in
Table B is a “0”?
A: 3 of 50, or 0.06
Q: Shouldn’t it be 0.10?
A: No. The run is too short to determine
probability. (Probability is the proportion in
an
infinite
series.)
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Probability Models
Probability models consist of two parts:
1) Sample Space (S) = the set of all possible
outcomes of a random process.
2) Probabilities for each possible outcome in
sample space S are listed.
Probability Model “toss a fair coin”
S = {Head, Tail}
Pr(heads) = 0.5
Pr(tails) = 0.5
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Rules of Probability
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Rule 1 (Possible Probabilities)
Let A ≡ event A
0 ≤ Pr(A) ≤ 1
Probabilities are always between
0 and 1.
Examples:
Pr(A) = 0 means A never occurs
Pr(A) = 1 means A always occurs
Pr(A) = .25 means A occurs 25% of the time
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Rule 2 (Sample Space)
Let S ≡ the entire Sample Space
Pr(S) = 1
All probabilities in the sample
space together must sum to 1
exactly.
Example: Probability Model “toss a fair
coin”, shows that Pr(heads) + Pr(tails) =
0.5 + 0.5 = 1.0
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Rule 3 (Complements)
Let Ā ≡ the complement of event A
Pr(Ā) = 1 – Pr(A)
A complement of an event is its
opposite
For example:
Let A ≡ survival  then Ā ≡ death
If Pr(A) = 0.95, then
Pr(Ā) = 1 – 0.95 = 0.05
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Rule 4 (Disjoint events)
Events A and B are disjoint if they are
mutually exclusive. When events are
disjoint
Pr(A or B) = Pr(A) + Pr(B)
Age of mother at first birth
(A) under 20: 25%
(B) 20-24: 33%
Pr(B or C) = 33% + 42% = 75%
(C) 25+: 42%
}
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Discrete Random Variables
Discrete random variables address outcomes
that take on only discrete (integer) values
Example:
A couple wants three children.
Let X ≡ the number of girls they will have
This probability model is discrete:
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Continuous Random Variables
Continuous random variables form a continuum
of possible outcomes.
• Example Generate
random number
between 0 and 1 
infinite possibilities.
• To assign
probabilities for
continuous random
variables  density
models (recall Ch 3)
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This is the density model for random
numbers between 0 and 1
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Area Under Curve (AUC)
The AUC concept (Chapter 3) is essential to
working with continuous random variables.
Example: Select a
number between 0
and 1 at random.
Let X ≡ the random
value.
Pr(X < .5) = .5
Pr(X > 0.8) = .2
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Normal Density Curves
Introduced in Ch 3: X~N(µ, ).
♀ Height
X~N(64.5, 2.5)
→
z
x
Standardized
Z~N(0, 1)

z
x
Z Scores
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68-95-99.7 Rule
• Let X ≡ ♀ height
(inches)
• X ~ N (64.5, 2.5)
• Use 68-95-99.7 rule
to determine heights
for 99.7% of ♀
• μ ± 3σ
= 64.5 ± 3(2.5)
= 64.5 ± 7.5
= 57 to 72
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If I select a woman at
random  a 99.7% chance
she is between 57" and 72"
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Calculating Normal Probabilities
when 68-95-99.7 rule does not apply
Recall 4 step procedure (Ch 3)
A: State
B: Standardize
C: Sketch
D: Table A
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Illustration: Normal Probabilities
What is the probability a woman is between 68” and
70” tall? Recall X ~ N (64.5, 2.5)
A: State: We are looking for Pr(68 < X < 70)
B: Standardize
(68  64.5)
z
 1.4
2.5
(70  64.5)
z
 2.2
2.5
Thus, Pr(68 < X < 70) = Pr(1.4 < Z < 2.2)
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Illustration
(cont.)
C: Sketch
D: Table A:
Pr(1.4 < Z < 2.2)
= Pr(Z < 2.2) − Pr(Z < 1.4)
= 0.9861 − 0.9192
= 0.0669
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