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Introduction to Probability II Eleisa Heron Neuropsychiatric Genetics Research Group Trinity College Dublin 22/10/08 Review – Probability I – Origins of probability – 17th C. gambling, frequentist and Bayesian interpretations – Definitions – trial, outcome, event, sample space – Probability Rules – [0,1], impossible and certain events – Mutually/Non-Mutually Exclusive Events – OR means add rule (ME) Venn diagrams – Conditional Probability – conditional probability fallacy 22/10/08 Probability II E. Heron 2 Bayes’ Theorem • Conditional probability for event • Using this we can say • Giving Bayes’ theorem 22/10/08 Probability II given that event E. Heron has occurred 3 Bayes’ Theorem and Bayesian Inference • Sometimes, Bayes’ theorem is written in the following way • In Bayesian inference, given data and parameters want to estimate the parameters 22/10/08 Probability II E. Heron of a model, we 4 Bayes’ Theorem Example • Court Setting Suppose a juror wants to combine DNA evidence that is presented to him/her together with their own prior beliefs in order to decide if defendant is guilty or not Evidence: DNA that matches the defendant’s DNA was found at the crime scene • • • is the event the defendant is guilty is the event the defendant is not guilty i is the event that the defendant’s DNA matches the DNA found at the crime scene • Interested in • Probability of being guilty before any evidence is presented • Probability of the event 22/10/08 Probability II E. Heron 5 Bayes’ Theorem Example • From Bayes’ theorem we have • Suppose a male committed the crime in a town with a male population of 20,000 • Juror uses this for his prior • This gives 22/10/08 Probability II E. Heron 6 Heads and tails – 0’s and 1’s 10110011001111111110 11010110000111111010 00100111000010000001 00101111001001000011 10110100110000110010 • 22/10/08 1010010100101001011 0 0101011011010101101 0 0010010101001011010 1 1010101100101001011 0 0100110101001011001 0 Which sequence have I made up and which is a true simulation? Probability II E. Heron 7 Boys and Girls • Five children in a family, assume probability of a boy (B) = probability of a girl (G) = 0.5 GGGGG BBGBG BBBBB Which has higher probability? 22/10/08 Probability II E. Heron 8 Independent Events • Two events and are independent if • Toss a coin and get a head Toss the coin again Probability of getting a head again is independent of the first toss and the fact that we got a head 22/10/08 Probability II E. Heron 9 Independent Events • For independent events and then AND means MULTIPLY for independent events • Toss two coins 22/10/08 Probability II E. Heron 10 Independent Events • Can mutually exclusive events be independent? Two events A and B Mutually exclusive Not mutually exclusive Independent • Dependent Assuming A and B are non-trivial events 22/10/08 Probability II E. Heron 11 Marginal Probability • . the marginal probability of an event probability of the event • The probability of the event regardless of what other events occurred • Disease screening example 22/10/08 is the unconditional Positive Negative Disease P(Dis, Pos) P(Dis, Neg) P(Disease) Well P(Well, Pos) P(Well, Neg) P(Well) P(Positive) P(Negative) Probability II E. Heron 12 Odds • Odds of an event are defined as the ratio of the probability that the event will occur to the probability that the event will not occur • To convert from odds to probability • Probability more intuitive, but odds used in gambling and in analysis of binary outcome variables (logistic regression) for example 22/10/08 Probability II E. Heron 13 Odds Cont’d • Odds lie between 22/10/08 and , since probability lies between Probability II E. Heron and 14 Odds Ratio • Odds ratio (OR) is a measure of effect size • OR defined as the ratio of the odds of an event occurring in one group to the odds of the event occurring in another group is the probability of the event in the first group is the probability of the event in the second group OR =1: the condition or event is equally likely in both groups OR >1: the condition or event is more likely in the first group OR <1: the condition or event is less likely in the first group • Again, just like odds, OR’s lie between 22/10/08 Probability II and E. Heron 15 Pink and Blue Cards • Pick a card – What is the probability card is blue on both sides? • Pick a card, card is blue on one side – What is the probability card is blue on the other side? 22/10/08 Probability II E. Heron 16 Pink and Blue Cards Cont’d Card Picked Pink Pink 1/3 1/3 First Second 1/2 Pink Pink 1/2 Pink Pink 1/2 Pink Blue 1/2 Blue Pink 1/2 Blue Blue 1/2 Blue Blue Pink Blue 1/3 Blue Blue 22/10/08 Probability II E. Heron 17 Pink and Blue Cards Cont’d • is the event that the first card is blue and second card is blue 22/10/08 Probability II E. Heron is the event that the 18 Goats and Car – Monty Hall Problem • Game Show – 3 doors, a car behind one and a goat behind each of the others – Contestant picks one of the 3 doors – Game show host (who knows which door conceals the car) opens one of the remaining two doors to reveal a goat – Contestant offered the chance to swap the door they originally chose for the remaining door Should the contestant swap? 22/10/08 Probability II E. Heron 19 Goats and Car – Monty Hall Problem Cont’d Car location Host Opens 1/3 Door 2 1/3 Door 3 Don’t Swap Swap 1/2 Door 2 1/6 Car Goat 1/2 Door 3 1/6 Car Goat Door 3 1/3 Goat Car Door 2 1/3 Goat Car Door 1 1/3 Prob. Total 1 1 Assume Door 1 is chosen 22/10/08 Probability II E. Heron 20 Goats and Car – Monty Hall Problem Cont’d Assume contestant chooses door 1 and the presenter opens door 3 is the event that the presenter opens door 3 is the event that the car is behind door 1, similarly 22/10/08 Probability II E. Heron and 21 Goats and Car – Monty Hall Problem Cont’d 22/10/08 Probability II E. Heron 22 References • Most introductory statistics books have a probability section at the start • Statistics for Technology by C. Chatfield Basic introductory probability • Introduction to Probability by Charles M. Grinstead, J. Laurie Snell (1997) More in depth • Weighing the Odds: A Course in Probability and Statistics by D. Williams (2001) More advanced, looking at both frequentist and Bayesian approaches 22/10/08 Probability II E. Heron 23