Module 2 Presentation

Module 2 Presentation
Probability Refresher
Interpretations of Probability

Long-term frequency of repeated trials
(Frequentist/Classical School)

Expectation from logical or physical model



flipping a coin, rolling fair dice, selecting a card from a
standard deck
Number of sunny days in August in Durham, NC
Degree of Belief (Bayesian School)
Probability that Duke will win the NCAA championship
in 2011
 Probability that an oil spill will occur in the Gulf of
Mexico in the next decade.

Probability of an Event

The probability of any event, A, must be a number
between 0 and 1.


0 ≤ P(A) ≤ 1
Sometimes probabilities are expressed as fractions or
percents:




The probability a pregnant woman will have a boy is 0.51.
[P(A)=0.51]
The probability of rolling a 12 on two fair dice is 1/36.
[P(A)=0.0278]
The chances of rain tomorrow are 40%. [P(A)=0.40]
The probability of event A not occurring (written as Ā
or Ac and called the complement of A):

P(Ā) = 1 - P(A)


The probability a couple will have a girl is 0.49.
The probability of no rain tomorrow is 0.60.
Probability of Two Events (1)


Probability of one event occurring (e.g., A) is known
as a marginal probability.
Probability of two events (e.g., A, B) occurring
together is known as a joint probability.


P(A ∩B) = P(A & B) = P(A,B)
= “probability that A and B occur together”
= “joint probability of A and B”
Probability of one event occurring (e.g., A) given that
another event has occurred (e.g., B) is known as a
conditional probability.

P(A|B)
= “probability of A given that B has occurred”
= “conditional probability of A given B”
Probability of Two Events (2)

Probability of at least one of events A and B
occurring
 P(A

U B) = P(A) + P(B) – P(A,B)
Independence: Events A and B are
independent if the occurrence of B has no
impact on the probability that A occurs.
A
and B independent if P(A|B) = P(A)
 If A and B are independent, P(A,B) = P(A)∙P(B)
 IMPORTANT!: Dependence does not imply
causality.

Bayes Rule:
Example




The probability that a freshman will make a B or lower in
calculus is 0.85.
The probability that a freshman will make an A in chemistry
given that she makes an A in calculus is 0.70.
The probability that a freshman will make an A in calculus
given that she makes an A in chemistry is 0.47.
Calculate the following:
1.
2.
3.
4.
The probability that a student will make an A in calculus.
The joint probability that a student will make an A in both
chemistry and calculus.
The marginal probability that a student will make an A in
chemistry.
The probability that a student will make an A in at least one of
the two classes.
Contingency Table
joint
B
A
A
p(A,B)
p(A,B)
p(B)
marginal
B
p(A,B)
p(A,B)
p(B)
p(A)
p(A)
1.0
marginal
Random Variables

Consider an “experiment”, the result of which is random. For example:






All of the above are considered random variables since we do not know
their values in advance.
X and Y are known as discrete random variables since their values only
take on particular values:



Flipping two coins and recording the number of tails, X.
Tossing two dice and recording the sum of both faces, Y.
Choosing a random adult male and recording his height, Z.
Choosing a random car in a parking lot and recording the fraction of fuel
remaining in the tank, V.
X = 0, 1, or 2
Y = any integer from 2 to 12
Z and V are known as continuous random variables since their values
can take on any number in a particular interval:


Z = any number > 0 [although values below and above certain thresholds are
highly unlikely]
V = any number between 0 (empty tank) and 1 (full tank)
Probability Distributions
Discrete Distribution




Probability Mass Function
(PMF)
Indicates the probability of
obtaining a value at each
possible point.
For the coin tossing example:
x
P(X = x)
0
0.25
1
0.50
2
0.25
All probabilities must sum to 1.
Probability Distributions
Continuous Distribution






Probability Density Function
(PDF)
Probability of any particular
number is zero.
Probability of obtaining a
range of values is indicated
by the area under the curve
between two values.
The total area under the
curve is 1.
PDFs are expressed in
equation form as opposed to
table form.
For the gas tank example one
PDF could be written:
Cumulative Distribution
Functions (1)

Cumulative Distribution Functions (CDFs)
indicate the probability that a random variable
takes on a value less than or equal to that
indicated:

For discrete variables, the CDF is a step
function that can be written in tabular form. For
x
P(X ≤ x)
our coin tossing example:
0
0.25
1
0.75
2
1.00
Cumulative Distribution
Functions (2)

For continuous variables, the CDF is smooth
and is typically given by an equation. For our
gas tank example (don’t worry about the
notation):

The utility of the CDF is that it allows you to
calculate the probability that a random variable
will be in a range of values by subtracting:
Cumulative Distribution
Functions (3)
Questions
What is the probability that a
random gas tank is between
40% and 60% full?
What is the probability that a
random gas tank is at least
40% full?
What is the probability that a
random gas tank is no more
than 60% full?
The Normal Distribution


A continuous distribution also known as a
Gaussian distribution or a “bell-shaped” curve.
Shape is described by two parameters:
 Mean
= “average value” = often symbolized by μ
 Variance = “spread” = often symbolized by σ2

Probably the most commonly encountered
distribution because of its connection to the
Central Limit Theorem which states (simply):
The mean of a sample drawn from any
distribution approaches a Normal distribution
as the sample size increases.
The beginning…
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