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Introduction to Probability I Eleisa Heron Neuropsychiatric Genetics Research Group Trinity College Dublin 15/10/08 Introduction “Probability and statistics used to be married; then they separated; then they got divorced; now they hardly ever see each other.” D. Williams, Weighing the Odds (2001) 15/10/08 Probability I E. Heron 2 Introduction Statistical Analysis Descriptive Statistics Statistical Inference • Descriptive Statistics: organising, presenting and summarising data • Statistical inference: observe a sample from a population and want to infer something about the population - need probability! 15/10/08 Probability I E. Heron 3 Outline • 2 lectures: – – – – – – Origins of probability, frequentist, Bayesian Definitions Probability Rules Mutually Exclusive Events Non-Mutually Exclusive Events Conditional Probability – – – – – – Bayes’ Theorem Independent Events Marginal Probability Odds Fun Problems! References 15/10/08 Probability I E. Heron 4 Origins of Probability • • We all have an intuitive idea of what probability is Can think of probability as a mathematical expression of the relationship between a particular outcome of an experiment and the total number of possible outcomes: toss a coin: roll a dice: (assumptions?) • Modern probability – gambling 17th Century – mathematicians Blaise Pascal and Pierre Fermat worked out the theory of probabilities in response to a gambling problem Many others along the way … • Foundations of probability 1930’s Russian mathematician A.N. Kolmogorov (measure theory, branch of pure mathematics) 15/10/08 Probability I E. Heron 5 Probability Interpretations – Frequentist vs Bayesian • Most use probability unthinkingly in a “frequentist way” • Frequentist - probability of an event is given by the fraction of times the event occurs on average over a large number of trials/experiments – Approximation to true probability – Improves as number of trials increases – Assumption that trial can be repeated many times (always valid? – winning an election, passing a test ??) • Bayesian (subjective) probability - reflects a person’s opinion about how likely an event is to occur, it represents the person’s strength of belief – Prior beliefs are updated using observed data – Valid in situations where long run relative frequency is not applicable 15/10/08 Probability I E. Heron 6 Probability Interpretations – Frequentist vs Bayesian What’s the correct approach? • Long run relative frequency idea of probability: intuitive motivation • Long run relative frequency can’t be made mathematically rigorous • Measure theory takes care of this problem • Once the rules of probability are defined – individual’s choice on interpretation as frequentist or Bayesian 15/10/08 Probability I E. Heron 7 Definitions Trial any process, which when repeated, generates a set of results or observations (trial = experiment) eg. Tossing a coin, rolling a dice Outcome the result of carrying out a trial eg. A head, a 4 Event a set consisting of one or more of the possible outcomes of a trial eg. The event getting a head, the event getting a 4 or a 6 Sample Space the set of all possible outcomes of a trial eg. Toss a coin – head and tail Throw a dice – 1, 2, 3, 4, 5, and 6 15/10/08 Probability I E. Heron 8 Probability Rules 1. For an event 2. 3. is defined to be the event not occurring If an event cannot occur: if an event will definitely occur: 15/10/08 Probability I (impossible event), (certain event) E. Heron 9 Mutually Exclusive Events – Addition Rule • Two or more events are said to be mutually exclusive if the events cannot occur simultaneously • For mutually exclusive events • Sum of probabilities of all mutually exclusive events is 1 OR means ADD for mutually exclusive events 15/10/08 Probability I E. Heron 10 Mutually Exclusive Events – Example Example: A card is dealt from a pack of well-shuffled cards. What is the probability that the card is either a heart or a spade? 15/10/08 Hearts Clubs Diamonds Spades Probability I E. Heron 11 Example: Toss a Single Coin Getting a head A head not occurring Getting 5 heads Getting a head or a tail 15/10/08 Probability I E. Heron 12 Example: Roll a Dice Getting a 1 Getting a 4 Getting a number greater than 1 OR 15/10/08 Probability I E. Heron 13 Non-Mutually Exclusive Events • Mutually Exclusive? – The events getting a head on tossing a coin and getting a tail? – The events getting a six and getting a three when throwing a dice? – The events getting an ace and getting a diamond when we pick a card at random? • If two events and are not mutually exclusive, can both happen together: • Example: Group of 10 people, 5 taking drug A, 3 people taking drug B, 2 people taking drug A and drug B and 4 people taking no drug, choose a person at random – Probability the person takes drug A or drug B? 15/10/08 Probability I E. Heron 14 Non-Mutually Exclusive Events Drug A Drug B 3 2 1 4 Example: Group of 10 people, 5 taking drug A, 3 people taking drug B, 2 people taking drug A and drug B and 4 people taking no drug, choose a person at random – Probability the person takes drug A or drug B? 15/10/08 Probability I E. Heron 15 Conditional Probability • Two events: • Example: Group of 10 people, 5 taking drug A, 3 people taking drug B, 2 people taking drug A and drug B and 4 people taking no drug, choose a person at random and , occurring, given that event 15/10/08 is the conditional probability of event occurred Probability I E. Heron 16 Conditional Probability Fallacy • Explore difference between • Example: and – For a particular disease, 1% of the population suffers from the disease – Disease is curable if detected early – Screening test for the disease is 99% accurate (99% sensitivity and specificity) Want to explore the idea of screening people for the disease, but want to think about the distress that would be caused to someone who doesn’t have the disease but is incorrectly found to have the disease by the screening test 15/10/08 Probability I E. Heron 17 Conditional Probability Fallacy Cont’d • Example: – For a particular disease, 1% of the population suffers from the disease – Screening test for the disease is 99% accurate (99% sensitivity and specificity) (sensitivity) (specificity) 15/10/08 Probability I E. Heron 18 Conditional Probability Fallacy Cont’d • Have • Can work out: 15/10/08 Probability I E. Heron 19 Conditional Probability Fallacy Cont’d • What is the probability of having the disease GIVEN that the test gave a positive result? • Surprising result? • 50% of positives would be false positives – unacceptable • Difference between the probability of testing positive given that the person has the disease (0.99) and the probability of having the disease given that the person tests positive (0.5) 15/10/08 Probability I E. Heron 20