Probability Concepts and Formal Probability

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Probability Concepts and Formal Probability

• Presentation to
• Risk Analysis Training Workshop
• Hyderabad, India July 2007
• Rob McDowell, Sr. Staff Economist
• Risk Analysis Systems
• USDA-APHIS
• Riverdale MD USA

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Objectives

• Probability
• kinds of probability
• useful working model or definition of
  probability
• probability axioms and laws
• how we represent probability mathematically and
  graphically
• what information is contained in these
  representations and how to extract it
• how we use probability to determine likelihood
  of events and how to use it to characterize
  uncertainty

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Objectives, cont.

• What is a random variable?
• What is a probability distribution?
• What are the essential properties?
• How do we represent these, use them to represent
  uncertainty and/or variability
• Kinds of distributions?
• Some important distributions
• Mastery of the binomial distribution
• How we use these in probabilistic risk analysis

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Questionnaire/Quiz

• 1. Have you taken formal course in statistics
  and probability at college level? (yes/no)
• 2. If so, how many years since last formal class
  in statistics or probability?
• Problems
• a. I have an unfair coin. Probability of heads
  is 3/8. What is probability of tails?
• b. With this coin, what is probability of
  obtaining on three sequential coin tosses the
  following sequence T H H (H head,
  T tails)
• c. With this coin, what is the probability of
  obtaining two heads on three coin tosses,
  regardless of sequence?
• d. I toss this coin ten times and get 10 heads.
  What is probability of head on 11th toss?
• e. I have standard deck of cards 52 in total,
  four suites (hearts, spades, diamonds, and
  clubs). Each suite has 2,3,4,5,6,7,8,9,10,
  J,K,Q,A. I will draw three cards in
  succession, not replacing the cards to the deck
  after each draw. What is probability of obtaining
  the cards in this sequence?
• 1st 2nd 3rd
• K of Hearts any 4 Q of Spades or any
  5

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A Little Background

• Probability theory relatively new branch of
  mathematics (geometry goes back to Greeks
  probability theory began with Pascal and others).
• Welcome to 20th century explosion of
  probabilistic thinking quantum mechanics showed
  that the physical world is characterized by
  uncertainty down to most fundamental parts of the
  universe.

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Background, continue.

• Profoundly unsettling to some Albert
  Einstein I dont believe God plays dice with
  the universe.
• Not so to others
• Neils Bohr Stop telling God what to do!

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Why Bother to Master the Concepts of Probability?

• A good foundation in the basic ideas of
  probability and statistics is absolutely
  essential to the adequate treatment of
  uncertainty in most quantitative policy analyses
  including risk assessments.
• Mitchell Small, Chapter 5, Probability
  Distributions and Statistical Estimation, in
  Uncertainty A Guide to Dealing with Uncertainty
  in Quantitative Risk and Policy Analysis by M.
  Granger Morgan, Max Henrion, and Mitchell Small.
  Cambridge Univ. Press. 1990.

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Why Bother to Master the Concepts of Probability?

• A good foundation in the basic ideas of
  probability and statistics is absolutely
  essential to the adequate treatment of
  uncertainty in most quantitative policy analyses
  including risk assessments.
• Mitchell Small, Chapter 5, Probability
  Distributions and Statistical Estimation, in
  Uncertainty A Guide to Dealing with Uncertainty
  in Quantitative Risk and Policy Analysis by M.
  Granger Morgan, Max Henrion, and Mitchell Small.
  Cambridge Univ. Press. 1990.

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What is a Random Variable?

• Anything that shows variation or uncertainty.
• What is a Probability Distribution
• Anything that assigns a number to each and
  every element of the random variable such that
  those numbers correspond to axioms of probability.

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Probability Distributions, recap

• Continuous means the random variable can hold
  infinitely many values (regardless of domain of
  r.v.)
• Discrete random variable can hold only integer
  values.
• Two functions describe RVs
• Distribution function F(x) Pr(X
• Probability density function f(x) F(x)
  (cont)
• Probability mass function P(xparameters)

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Probability Concepts and Axioms

• Presentation to
• Risk Analysis Training Workshop
• Hyderabad, India July 2007
• Risk Analysis Systems
• USDA-APHIS-PPD
• 4700 River Road
• Riverdale MD 20737 USA
• Created in Microsoft PowerPoint 2002

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The Problem With Probability

• There is no completely satisfactory
  definition of probability.
• Probability is one of those elusive concepts
  that virtually everyone knows but which is nearly
  impossible to define entirely adequately.
• Theodore Colton, Statistics in Medicine,
  Chp.3. Little, Brown, and Co. 1974,

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Probability Concepts

• Counting Definition (historically important)
• Pr(A) number of ways A can occur
• total no. of possible outcomes
• Restricted to situations where
• outcomes are discrete events
• all outcomes equally likely
• (rolling dice, card draws, etc.)
• Quite cumbersome in complex
  situations.

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Probability Concepts, cont.

• Relative Frequency Definition
• Pr(A) long-run relative frequency of event A.
• Pr(A) Lim x/n
• x ? 0 where
• x number of times event A occurs
• N number of trials or occasions when
  event A could occur.
• Limited to situations where n is large.

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Probability Concepts, cont.

• Subjective or Personal Probability
• Pr(A) personal statement of belief about
  the likelihood of event A occurring.
• Can be used when (1) event has never occurred,
  e.g. x is zero for Pr(A) x/n , or (2) n is
  zero, e.g. situation where A might occur has
  never occurred. Example Pr(crash on landing
  during first manned mission to Mars).
• DeFineitt remarked all probabilities are
  subjective.

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Probability Concepts, cont.

• Probability as Logical Concept
• Probability viewed not as personal belief but
  as the logical result given the evidence
  available.
• Pr(A) F(E) function of available evidence
• This leads to useful and widely used (in
  field of quantitative risk analysis) distinction
  between probability and frequency and a
  definition rooted in this distinction.

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Probability Concepts, cont.

• Probability as Logical Concept, cont.
• Frequency is empirical, it occurs in the
  real world, it can be counted, can be speculated
  upon in thought experiments.
• Probability is indication of degree of
  belief it exists in the mind, not in the outside
  world it is based on the evidence available to
  the individual.
• The adverse consequences of unwanted events are
  not caused by the probabilities of those events
  they are caused by events and the frequency with
  which they occur. A probability never hurt
  anyone.

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Probability Concepts, cont.

• Probability as Logical Concept, cont.
• Powerful and useful definition is that
  developed by Kaplan
• probability of frequency definition
  likelihood is expressed as probability (our
  degree of belief based on the evidence) of a
  particular value for the frequency with which an
  event or hypothesis holds.
• This has become the standard definition of
  probability in the reliability engineering field
  and is at the heart of modern risk assessment.
• George Apostolakis, Science ..

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Probability Axioms (20th Century)

• A probability is a real number between 0 and 1,
  inclusive 0
• Sample space S complete set of all possible
  outcomes E1,E2,E3,En
• P(S) 1.0
• Probability of ith Event denoted P(Ei) or Pr(Ei)
• ? P(Ei) 1.0 when summed over all i when
  E1,E2,mutually exclusive.

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Probability Axioms Laws, cont.

• Given a sample space S A,B,C
• and Pr(A),
• the compliment of A everything in S
  except A and is denoted Å. (not A)
• Thus P(Å) P(S) P(A) 1 P(A)

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Probability Axioms Laws, cont.

• Set Operations and Probability
• For S A,B,C and A,B,Cnot mutually
  exclusive
• A n B (read A intersection B) is defined
  as both A and B occurring.
• A U B (read A union B) is defined as
• A or B or both A and B occurring.

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Probability Axioms Laws, cont.

• Computing Probability of Intersection and Union
• P(A n B) P(A) P(BA)
• where P(BA) is the conditional probability
  of B occurring, given that A has occurred.
• If P(BA) Pr(B) we say A and B are
  independent
• and thus P(A n B) P(A)P(B)
• Sometimes called the multiplicative law of
  probability.
• P(A U B) P(A) P(B) P(A n B)
• P(A) P(B) P(A)P(BA)
• If A and B independent P(A) P(B) P(A)P(B)
• Sometimes called the additive law of probability.

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Probability Axioms Laws, cont.

• Conditional Probability One of the most
  important concepts for risk and reliability
  modeling, quantitative epidemiology, etc.
• From the multiplicative law Pr(A n B)
  Pr(A) Pr(BA)
• re-written
  P(AB) P(A)P(BA)
• we obtain the formal definition of conditional
  probability
• P(BA) Pr(A n B) / P(A) or P(AB)
  / P(A)
• What does P(AB) 0 mean?
• This means A and B are mutually exclusive or
  disjoint.
• Exercise P(A) 0.3, P(B0.4), P(BA) 0.
• Compute P(A n B)

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Probability Axioms Laws, cont.

• DeMorgans Laws
• An(B U C) (AnB) U (CnA)
• A U (B n C) (A U B) n (A U C)
• compliment (A U B) (compliment A) n
  (compliment B)
• compliment (A n B) (compliment A) U
  (compliment B)
• In general do NOT assume operational laws in
  numerical algebra apply to algebra of sets. Set
  operations are governed by laws of Boolean
  algebra.