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

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).

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)

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

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

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 {outcome 1, outcome 2,…} The probability of outcome 1 is written P(outcome 1 )

Heads or Tails (fair coin) OutcomeProbability Heads0.5 Tails0.5

In the play Rosenkrantz and Guilderstein are Dead Outcome (note same sample space) Probability Heads1.0 Tails0.0

Fair Die Outcome Prob1/6

A Pair of Fair Dice /362/363/364/365/366/365/364/363/362/361/36

Can You Imagine a Way of Testing Whether a Die is Fair? Discuss

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 outcome 1 is p. The total probability of any other outcome is 1.0-p.

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?

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) = P(E)

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

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)

Bayes Theorem Suppose you know P(A|B) and you want to calculate P(B|A) –P(B|A)P(A) = P(A|B)P(B) –P(B|A) = P(A|B)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 don’t. You just had a positive test result. What are your chances?

Solution Look at people, of whom 1000 have the disease – don’t; hence false positives –1000 do; hence 999 true positives –Your chances of having the disease given you had a positive test result are 999/( ) = 1/21. Why? P(I and X) = P(I|X)P(X) = P(X|I)P(I) P(I|X) = P(X|I)P(I)/P(X) = 0.999*0.001/P(X) And P(X) = P(X|I)P(I)+P(X|not I)P(not I) = 0.999* *0.999 = 0.021*0.999 So P(I|X) = 1/21

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.

Demonstration Using R

Continuous Random Variables Random variables can be continuous –Your height –Your weight –Your age

Examples of Continuous Random Variables Using R

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.

Examples

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.