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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

  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.

Complement

Set Theory

Intersection

“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

  For any event A, P(AC) = 1 – P(A).

  Let denote the empty set. Then P( ) = 0.

  If A is an event, and A = {O1, O2, …, On}, then P(A) = P(O1) + P(O2) +….+ P(On).

  Addition Rule (for when A and B are not mutually exclusive):

A Few Useful Things

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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”

“The Multiplication Rule”

If two events are independent, the probability of occurrence of the intersection reduces to:

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

a collection of objects. The number of permutations of n objects is n!.

•  The number of permutations of k objects chosen from a group of n objects is n!/(n – k)!

•  When order matters, use permutations.

•  Combinations are an unordered collection of objects.

•  The number of combinations of k objects chosen from a group of n objects is:

n!/[(n – k)!k!].

•  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:

Union (mutually exclusive events):

Union (general):

Conditional Probability:

Intersection (independent events):

Intersection (general):

Negation (complement):

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P(A∩ B) = P(A |B) ⋅P(B)