Advanced Mathematical Methods Part II Statistics Probability Mel - PowerPoint PPT Presentation

Advanced Mathematical Methods Part II – Statistics Probability Mel Slater http://www.cs.ucl.ac.uk/staff/m.slater/Teaching/Statistics/ Letters between these two in 17C about wagers led to foundations of probability theory Blaise Pascal Pierre de Fermat www.abarnett.demon.co.uk/atheism/wager.html 1

Outline � Probability � Axioms – Calculus of probability � Conditional Probability and Independence � Assigning Probabilities 2

Probability � A measure of the degree of belief in the truth of some proposition or the occurrence of some event. � Probability theory is concerned with how to find the probabilities of more complex events (propositions) given the probabilities of constituent simpler events. 3

Domain or Sample Space � Probabilities are defined over a domain of ‘elementary events’. � This is also called a sample space. � Complex (compound) events are formed out of the elementary events. � Mathematically the domain/sample space is a set. 4

Examples � Questionnaire (Virtual Reality Example): To what extent did you have a sense of being in the place depicted by the virtual reality? • Answer 1=Very Low … 7=Very High • Domain = {1,2,3,…,7} � Two coins tossed: {HH,HT,TH,TT} � Consider “It will rain tomorrow” – what is the domain? 5

Axioms of Probability � We use the term ‘this event is true’ to mean that it has or will happen, and ‘false’ otherwise. http://kolmogorov.com/ � P(E) means ‘probability of proposition E’. 6

Axioms of Probability - Events Examples – the questionnaire, the coin tossing 7

Axioms of Probability � The triple � … is the object of study: p is a probability measure, which satisfies a number of axioms: Such E i are called ‘mutually exclusive’ – one and only one can be true 8

Complements of Events � The event not-E is written as � It is easy to show from the axioms that 9

Conditional Probability � All probabilities are conditional probabilities. � p ( E|X ) is the probability of E given that we know X to be true. � Often the conditions are simply understood, in which case p ( E ) can be used. 10

Independence � Conditional probability is the final axiom: � This leads to the definition of independent events: A and B are independent if and only if: p(A|B) = p(A) and p(B|A) = p(B). � The complete definition for n events is: � Where K can be any non-empty subset of {1,2,…,n} 11

Probability of A or B � From the axioms the following can easily be proved: 12

Assigning Probabilities � Since probabilities are ‘subjective’ they can be assigned however you like. � But some choices are more rational than others. � Three methods are considered: • Equal probabilities • Betting odds • Frequency 13

Equal Probabilities � When there is no prior information over the domain, every elementary event in the domain should have equal probability. � This is the case in classic ‘wagers’ such as dice, coins, cards. � Eg, in 3 tosses of a coin the domain is • {HHH,HHT,HTH,THH, TTT,TTH,THT,HTT} � Each can be assigned probability 1/8. 14

Betting Odds � Suppose that the betting odds on E are a:b (or a/b:1). � This means that for every b pounds you bet you get a if E occurs. � If you accept these odds then you are agreeing that: � p(E) = b/(a+b). � Alternatively p(not-E)/p(E) = a/b 15

Fair Betting Odds � Betting odds are considered ‘fair’ when each party expects on the average to gain 0. • Suppose odds of x:1 are agreed for event E, then p(E) = 1/(x+1). • Suppose that probability reflects long run frequency – i.e., that E happens 1/(x+1) of the time. • Then 1/(x+1) of the time you will win x pounds and x/(x+1) of the time you will lose 1 pound. • Therefore your long run expected gain is x/(x+1) – x/(x+1) = 0. 16

Probability as Frequency � Suppose there are repeated trials of an ‘experiment’ in which E is a possible outcome. � The trials are all under equal conditions and independent of one another. � Suppose the number of trials is n, and E happens r times. Then � r/n → p(E) as n →∞ (with prob. 1) 17

Summary � Probability measures degree of belief over a domain. � The events or propositions that are elements of the domain are elementary events. � However probabilities are assigned to these elementary events the calculus of probability applies. � This is based on the axioms of probability � Some terms to remember: mutually exclusive, conditional, independence. 18