ST 380
Probability and Statistics for the Physical Sciences
What is Probability?
Probability (or likelihood) is a measure or estimation of how likely it
is that something will happen or that a statement is true.a
a
http://en.wikipedia.org/wiki/Probability
Probability is an intuitive concept, but we need a mathematical
framework to make precise calculations.
Do not let the mathematical framework destroy your intuition!
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Probability
Introduction
ST 380
Probability and Statistics for the Physical Sciences
Sample Space
Experiment
An experiment is an activity whose outcome is unknown in advance.
For example, randomly choose a North Carolina voter and ask some
questions.
Sample space
The sample space is the set of all possible outcomes; notation: S.
In the example, the list of possible voter responses.
If the only question is “Do you approve of Hillary Clinton”, the
sample space could be S = {“yes”, “no”, “no opinion”}.
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Probability
Sample Spaces and Events
ST 380
Probability and Statistics for the Physical Sciences
Event
An event associated with an experiment is something that either
occurs or does not occur, and we know which when we know the
outcome of the experiment.
We can identify the event with the set of outcomes for which it
occurs; that is, an event is effectively a subset of the sample space;
notation: A ⊆ S.
If A is “voter approves of Clinton”, then A = {“yes”}.
Simple Event
An event like this that contains just one outcome is a simple event. It
is essentially the same as the outcome itself.
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Probability
Sample Spaces and Events
ST 380
Probability and Statistics for the Physical Sciences
Some Set Theory
Complement
The complement A0 of an event A is the set of all outcomes not in A;
think of it as “not A”.
If A is “voter approves of Clinton”, then A0 = {“no”, “no opinion”}.
Union
The union A ∪ B of two events A and B is the set of all outcomes in
either A or B or both; “A or B”.
Intersection
The intersection A ∩ B of two events A and B is the set of all
outcomes in both A and B; “A and B”.
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Probability
Sample Spaces and Events
ST 380
Probability and Statistics for the Physical Sciences
Null Event
What if A and B have no outcomes in common?
The null event, or empty set, is denoted ∅.
Mutually Exclusive Events
When A and B have no outcomes in common, that is A ∩ B = ∅,
they are mutually exclusive.
If A is “voter approves of Clinton”, and If B is “voter does not
approve of Clinton”, then A and B are mutually exclusive.
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Probability
Sample Spaces and Events
ST 380
Probability and Statistics for the Physical Sciences
Axioms
The axioms of a theory are the minimum set of properties from which
all other properties can be derived.
Non-negativity
For any event A, the probability of A, P(A), satisfies P(A) ≥ 0.
Sure Event
P(S) = 1.
Countable Additivity
If A1 , A2 , . . . is an infinite collection of mutually exclusive events, then
P (A1 ∪ A2 ∪ . . . ) =
∞
X
P(Ai ).
i=1
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Axioms, Interpretations, and
Probability Properties of Probability
ST 380
Probability and Statistics for the Physical Sciences
Derived Properties
Null Event
P(∅) = 0.
Complementary Events
P(A0 ) = 1 − P(A).
Finite Additivity
If A1 , A2 , . . . , Ak is a finite collection of mutually exclusive events,
then
k
X
P (A1 ∪ A2 ∪ · · · ∪ Ak ) =
P(Ai ).
i=1
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Axioms, Interpretations, and
Probability Properties of Probability
ST 380
Probability and Statistics for the Physical Sciences
Interpreting Probability
Repeated Trials
Suppose that the experiment can be carried out repeatedly in the
same way.
For example, we could contact many voters.
For some event A, we see n(A) occurrences after n trials.
Relative Frequency
The relative frequency of A is n(A)/n.
After many trials, we expect the relative frequency of A to be close
to the probability P(A).
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Axioms, Interpretations, and
Probability Properties of Probability
ST 380
Probability and Statistics for the Physical Sciences
Equally Likely Outcomes
In some experiments, we may assume that all outcomes are equally
likely, at least as an approximation.
Coin tossing
If a coin is tossed, the sample space is S = {”heads”, ”tails”}. If the
coin is “fair”, both outcomes are equally likely, so the probability of
each simple event is 1/2.
No coin is exactly fair, but it is a good approximation.
In general, if the experiment has N outcomes and they are equally
likely, the probability of each simple event must be 1/N.
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Axioms, Interpretations, and
Probability Properties of Probability
ST 380
Probability and Statistics for the Physical Sciences
Counting
If an experiment has N equally likely outcomes, and some event A
occurs for N(A) of them, then
P(A) =
N(A)
.
N
For this type of experiment, one way to calculate P(A) is by counting
the outcomes that are favorable to A.
The branch of mathematics that studies ways of counting various
things is known as combinatorics.
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Probability
Counting Techniques
ST 380
Probability and Statistics for the Physical Sciences
Common Birthdays
In a room of M = 40 people, what is the chance that two have the
same birthday?
The experiment consists of asking M people their birthdays, say as a
number from 1 (January 1) to 365 (December 31) (people born on
February 29th in leap years do not exist, for this calculation).
Each outcome is a list of M numbers from 1 to 365, so the sample
space S = {1, 2, . . . , 365}M , and N = 365M .
As an approximation, assume the outcomes are equally likely.
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Probability
Counting Techniques
ST 380
Probability and Statistics for the Physical Sciences
Instead of
A = “two or more people have the same birthday”,
we focus on
A0 = “no two people have the same birthday”,
because P(A) = 1 − P(A0 ), and P(A0 ) is easy to calculate.
For M = 2,
P(A0 ) =
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364
.
365
Probability
Counting Techniques
ST 380
Probability and Statistics for the Physical Sciences
In general,
P(A0 ) =
364 363
365 − (M − 1)
×
× ··· ×
.
365 365
365
For M = 40, P(A0 ) = 0.11, so P(A) = 0.89.
That is, there is an 89% chance that two people in this room have
the same birthday.
In gambling terms, the odds are around 8:1 that two people in this
room have the same birthday.
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Probability
Counting Techniques
ST 380
Probability and Statistics for the Physical Sciences
From the general formula,
P(A0 ) =
N(A0 )
.
N
In the expression for P(A0 ), multiply numerator and denominator by
365:
P(A0 ) =
365 364 363
365 − (M − 1)
×
×
× ··· ×
.
365 365 365
365
The denominator is 365M = N, so the numerator must be N(A0 ).
That is, we have calculated N(A0 ), and can get N(A) = N − N(A0 ),
without going through all N outcomes to decide which are favorable
to A. (FYI: N(A0 ) = 33.7 × 10100 = 33.7 googols.)
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Probability
Counting Techniques