Probability

1
Probability

• Harry R. Erwin, PhD
• School of Computing and Technology
• University of Sunderland

2
Resources

• Rowntree, D. (1981) Statistics Without Tears.
  Harmondsworth Penguin.
• Hinton, P.R. (1995) Statistics Explained. London
  Routledge.
• Hatch, E.M. and Farhady, H. (1982) Research
  Design And Statistics For Applied Linguistics.
  Rowley Mass. Newbury House.
• Crawley, MJ (2005) Statistics An Introduction
  Using R. Wiley.
• Gonick, L., and Woollcott Smith (1993) A Cartoon
  Guide to Statistics. HarperResource (for fun).

3
Module Outline

• Introduction
• Using R
• Data analysis (the gathering, display, and
  summary of data)
• Probability (the laws of chance)
• Statistical inference (the drawing of conclusions
  from specific data knowing probability)
• Experimental design and modeling (putting it all
  together)

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Lecture Outline

• Introduction
• Basic Definitions
• Basic Operations
• Conditional Probability
• Independence
• Bayes Theorem
• Discrete Random Variables
• Continuous Random Variables
• Examples

5
Introduction

• Historically, probability has had one
  application gambling
• Claudius (Roman emperor, 10 BCE-54 CE) wrote the
  first book on gambling How to win at dice
• Blaise Pascal and Pierre de Fermat invented the
  modern theory of probability

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Basic Definitions

• Random experiment the process of observing a
  chance event
• Elementary outcomes the possible results
• Sample space the collection of all elementary
  outcomes, written outcome1, outcome2,
• The probability of outcome1 is written P(outcome1)

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Heads or Tails (fair coin)
Outcome Probability
Heads 0.5
Tails 0.5
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In the play Rosenkrantz and Guilderstein are Dead
Outcome (note same sample space) Probability
Heads 1.0
Tails 0.0
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Fair Die
Outcome 1 2 3 4 5 6
Prob 1/6 1/6 1/6 1/6 1/6 1/6
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A Pair of Fair Dice
2 3 4 5 6 7 8 9 10 11 12
1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
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Can You Imagine a Way of Testing Whether a Die is
Fair?

• Discuss

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Rules of Elementary Probability

• The probability of any outcome in the sample
  space is between 0.0 and 1.0 (non-negative)
• The total probability of all outcomes in the
  sample space is 1.0
• Suppose the probability of outcome1 is p. The
  total probability of any other outcome is 1.0-p.

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Basic Operations

• An event is a set of elementary outcomes. The
  probability of the event is the sum of the
  probabilities of the outcomes in the set.
• An event is written list of outcomes
• Suppose you roll a die twice, and the sum is 3.
  The set of events corresponding to this is
  (1,2),(2,1)
• The probability of this event is 1/36 1/36
• What is the probability of getting a sum of 4?

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Possible Events

• E and F, meaning both event E and event F
  occur.
• E or F, meaning either event E or event F
  occur.
• not E, meaning event E does not occur.
• P(E or F) P(E) P(F) - P(E and F)
• If the events are mutually exclusive, P(E or F)
  P(E) P(F)
• Finally, P(not E) 1.0 - P(E)

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Conditional Probability

• Suppose you have two events E in sample space A
  and F in sample space B.
• P(EF) is the probability of E given that F
  happens.
• P(EF) P(E and F)/P(F)
• P(E and F) P(EF)P(F) P(FE)P(E)
• Note P(EE) P(E and E)/P(E) P(E)/P(E) 1
• Also P(EF) 0 if they are mutually exclusive

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Independence

• Two events are independent if the occurrence of
  one has no influence on the other.
• If two events, E and F, are independent,
• P(E and F) P(E)P(F)

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Bayes Theorem

• Suppose you know P(AB) and you want to calculate
  P(BA)
• P(BA)P(A) P(AB)P(B)
• P(BA) P(AB)P(B)/P(A)
• A rare disease has a prevalence of 1/1000
• There is a test that is 99 accurate when you
  have the disease.
• The test also reports 2 positives when you
  dont.
• You just had a positive test result. What are
  your chances?

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Solution

• Look at 1000000 people, of whom 1000 have the
  disease
• 999000 dont hence 19980 false positives
• 1000 do hence 999 true positives
• Your chances of having the disease given you had
  a positive test result are 999/(19980999)
  1/21. Why?
• P(I and X) P(IX)P(X) P(XI)P(I)
• P(IX) P(XI)P(I)/P(X) 0.9990.001/P(X)
• And P(X) P(XI)P(I)P(Xnot I)P(not I)
  0.9990.0010.020.999 0.0210.999
• So P(IX) 1/21

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Discrete Random Variables

• A random variable is the numerical outcome of a
  random experiment.
• Each possible outcome has a probability.
• Histograms can be used to graph these.

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Demonstration Using R
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Continuous Random Variables

• Random variables can be continuous
• Your height
• Your weight
• Your age

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Examples of Continuous Random Variables Using R
23
You Can Also Discuss the Cumulative Probability
Distribution

• This the probability of a result between the
  smallest possible value and a given value.
• Mathematically, it is area, calculated by summing.

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Examples
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How you use probability

• In your experimental work, you show that a null
  hypothesis has very low probability of being
  correct.
• That means the necessary probabilities must be
  evaluated easily.
• Discussed in the next lecture.