CIVL 3103
Basic Laws and Axioms of
Probability
Why are we studying probability and
statistics?
•  How can we quantify risks of decisions
based on samples from a population?
•  How should samples be selected to
support good decisions?
Learning Objectives –
Basic Laws and Axioms of Probability
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Explain the basic laws and axioms of probability.
Describe the terms mutually exclusive and
independent, and explain their relevance.
Identify the appropriate method (i.e. union,
intersection, conditional, etc.) for solving a
problem.
Apply basic probability principles to solve
engineering-oriented problems.
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Probability vs. Statistics
•  Probability- parameters are known
from past history and we can deduce
behavior of system from a model.
•  Statistics-parameters are unknown and
must be estimated from available data
Random Experiment
A random experiment can result in different
outcomes every time it is repeated, even though the
experiment is always repeated in the same manner.
Ex. Call center
Basic Laws and Axioms of Probability
DEFINITIONS
•  Experiment – any action or process that generates
observations (e.g. flipping a coin)
•  Trial – a single instance of an experiment (one flip of the coin)
•  Outcome – the observation resulting from a trial (“heads”)
•  Sample Space – the set of all possible outcomes of an
experiment (“heads” or “tails”) (may be discrete or
continuous)
•  Event – a collection of one or more outcomes that share some
common trait
•  Mutually Exclusive Events – events (sets) that have no
outcomes in common.
•  Independent Events – events whose probability of occurrence
are unrelated
•  Null Set or Impossible Event – an empty set in the sample
space
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Venn Diagrams
Event A in sample space S.
Mutually exclusive events A and B.
Set Theory
Intersection
Complement
“outcomes in S contained in both A and B”
“outcomes in S not contained in A”
Set Theory
Union
“outcomes in S contained in either A or B or both”
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Definition of Probability
When conducting an experiment, the probability of
obtaining a specific outcome can be defined from
its relative frequency of occurrence:
Example: coin toss
Basic Axioms of Probability
• 
Let S be a sample space. Then P(S) = 1.
• 
For any event A,
• 
If A and B are mutually exclusive events, then
. More generally, if
are mutually exclusive events, then
.
A Few Useful Things
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For any event A, P(AC) = 1 – P(A).
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Let
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If A is an event, and A = {O1, O2, …, On}, then P(A) =
P(O1) + P(O2) +….+ P(On).
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Addition Rule (for when A and B are not mutually
exclusive):
denote the empty set. Then P(
) = 0.
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Examples
Orders for a certain type of lighting fixture have been summarized
according to the optional features that are requested for it:
no optional features = 0.3
one optional feature = 0.5
more than one option = 0.2
a.) What is the probability that an order includes at least one
optional feature?
b.) What is the probability that an order includes no more than
one optional feature?
Conditional Probability
The probability of A occurring given that B has already occurred:
The probability of occurrence of the intersection of two sets:
“Independent events”
If two events are independent, the probability of occurrence of the
intersection reduces to:
“The Multiplication Rule”
Examples
Oil wells drilled in region A have probability 0.2 of
producing. Wells drilled in region B have
probability 0.09 of producing. One well is drilled
in each region. Assume the wells produce
independently.
a)  What is the probability that both wells produce?
b)  What is the probability that neither well
produces?
c)  What is the probability that at least one of the
two produces?
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Examples
Fifteen of every 400 people is colorblind.
Fourteen of those are men and one is a woman.
Assume men make up half the population.
a.) What is the probability of being colorblind?
b.) What is the probability of being a colorblind
male?
c.) What is the probability of being colorblind IF
you are a male?
Counting Methods
•  A permutation is an ordering of •  Combinations are an unordered
a collection of objects. The
collection of objects.
number of permutations of n
objects is n!.
•  The number of combinations of
k objects chosen from a group
of n objects is:
•  The number of permutations of
k objects chosen from a group
of n objects is n!/(n – k)!
n!/[(n – k)!k!].
•  When order matters, use
permutations.
•  The number of ways to divide a
group of n objects into groups
of k1, … , kn objects where k1
+ … + kn = n, is:
n!/(k1!...kn!).
Examples
•  Ten engineers have applied for a management position in a
large firm. Four of them will be selected as finalists for the
position. In how many ways can this selection be made?
•  A chemical engineer is designing an experiment to determine
the effect of temperature, stirring rate, and type of catalyst
on the yield of a certain reaction. She wants to study five
different reaction temperatures, two different stirring rates,
and four different catalysts. If each run of the experiment
involves a choice of one temperature, one stirring rate, and
one catalyst, how many different runs are possible?
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Other Notation
Using this new shorthand, we can rewrite the basic axioms of probability as:
Negation (complement):
Union (mutually exclusive events):
Union (general):
Conditional Probability:
Intersection (independent events):
Intersection (general):
P(A ∩ B) = P(A | B) ⋅ P(B)
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