Introduction to probability theory Jouni Tuomisto THL

Probability of a red ball P(x|K) = R/N, –x=event that a red ball is picked –K=your knowledge about the situation

Probability of an event x If you are indifferent between decisions 1 and 2, then your probability of x is p=R/N. p 1-p Red x does not happen x happens White ball Decision 1 Red ball Decision 2 Prize 100 € 0 € 100 € 0 €

What is probability? – 1. Frequentists talk about probabilities only when dealing with experiments that are random and well- defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.[1] – 2. Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, or an objective degree of rational belief, given the evidence. –Source: Wikipedia

The meaning of uncertainty –Uncertainty is that which disappears when we become certain. –We become certain of a declarative sentence when (a) truth conditions exist and (b) the conditions for the value ‘true’ hold. –(Bedford and Cooke 2001) –Truth conditions: –It is possible to design a setting where it can be observed whether the truth conditions are met or not.

Different kinds of uncertainty –Aleatory (variability, irreducible) –Epistemic (reducible), actually the difference only depends on its purpose in a model. –Weights of individuals is aleatory if we are interested in each person, but epistemic if we are interested in a random person in the population. –Parameter (in a model): should be observable! –Model: several models can be treated as parameters in a meta-model

Different kinds of uncertainty: not really uncertainty –Ambiguity: not uncertainty but fuzziness of description –Volitional uncertainty: “The probability that I will clean up the basement next weekend.” –Uncertainties about own actions cannot be measured by probabilities.

Probability rules Rule 1 (convexity): –For all A and B, 0 ≤ P(A|B) ≤ 1 and P(A|A)=1. –Cromwell’s rule P(A|B)=1 if and only if A is a logical consequence of B. Rule 2 (addition): if A and B are exclusive, given C, –P(A U B|C) = P(A|C) + P(B|C). –P(A U B|C) = P(A|C) + P(B|C) – P(A ∩ B|C) if not exclusive. Rule 3 (multiplication): for all A, B, and C, –P(AB|C) = P(A|BC) P(B|C) Rule 4 (conglomerability): if {B n } is a partition, possibly infinite, of C and P(A|B n C)=k, the same value for all n, then P(A|C)=k.

Binomial distribution –You make n trials with success probability p. The number of successful trials k follows the binomial distribution. –Like drawing n balls (with replacement) from an urn and k being red. –P(n,k|p) = n!/k!/(n-k)! p k (1-p) (n-k)

Example –You draw randomly 3 balls from an urn with 40 red and 60 white balls. What is the probability distribution for the number of red balls?

Answer –P(n,k|p) = n!/k!/(n-k)! p k (1-p) (n-k) –0 red: 3!/0!/3! *0.4 0 *(1-0.4) 3-0 – = 1*1*0.6 3 = –1 red: 3!/1!/2! *0.4 1 *(1-0.4) 3-1 = –2 red: 3!/2!/2! *0.4 2 *(1-0.4) 3-2 = –3 red: 3!/3!/0! *0.4 3 *(1-0.4) 3-3 = 0.064

Binomial distribution (n=6)

Binomial distribution: likelihoods