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Today • Today: Some more counting examples; Start Chapter 2 • Important Sections from Chapter 1: 1.1-1.5; Please read 1.7-1.8 • Reading: – – – – Assignment #2 is up on the web site www.stat.lsa.umich.edu/~dbingham/stat405 Please read Chapter 2 Suggested problems: 2.4, 2.5, 2.7, 2.13, 2.25, 2.28, 2.32, 2R1, 2R2 Example • A batch of 50 manufactured items contains 5 defective items. If 10 items are randomly selected what is the probability that at least one of the defectives is found? Example • If there are 60 people in a Statistics 405 class, what is the probability that at least 2 have the same birthday? Partitions • A partition of an event F is a collection of events (E1, E2, …, Ek) that are mutually disjoint (i.e., EiEj= { }, for all i and j ) and where F E1 E2 ... Ek • Since none of the Ei’s have outcomes in common, then the number of outcomes in F is the same as the sum of the outcomes in each of the Ei’s • That is, if F has N outcomes and Ei has Ni outcomes, then N=N1+N2…+Nk Partitions • If E1, E2,…, Ek form a partition of F, then the number of distinguishable ways to partition N objects into k groups is N N! N1 , N 2 ,... N k N1!N 2 !... N k ! Example • What is the number of ways that 52 cards may be divided equally among 4 players Chapter 2 • Chapter 2 has two main topics: – – • Random variables and their distributions Conditional probability and independence Will begin with the second topic Conditional Probability Example: – – – – A family has two children The sample space for the possible gender of their children is {gg, gb, bg, bb}. The parents are seen leaving a Girl Scout meeting, suggesting that at least one of their children is a girl If each gender is equally likely, what is the probability that the other child is a boy Conditional Probability • • • Consider a probability model with sample space Let E be an event in the sample space If F is another event in the sample space where P(F)>0, the conditional probability of E given F is: P( E | F ) • P( EF ) P( F ) Idea: This represents the appropriate change in the probability assignment when we know that F has occurred Example • From a group of 5 Democrats, 5 Republicans and 5 Independents, a committee of size 3 is to selected • What is the probability that each group will be represented on the committee if the first person selected is an Independent? Some Useful Formulas • Multiplication Rule: P( EF ) P( E ) P( F | E ) P( F ) P( E | F ) • Law of Total Probability: P ( E ) P( E | F ) P ( F ) P ( E | F c ) P( F c ) • Bayes Theorem: P( F | E ) P( E | F ) P( F ) P( E | F ) P( F ) P( E | F c ) P( F c ) Where do these come from? Example • • • • Consider a routine diagnostic test for a rare disease Suppose that 0.1% of the population has the disease, and that when the disease is present the probability that the test indicates the disease is present is 0.99 Further suppose that when the disease is not present, the probability that the test indicates the disease is present is 0.10 For the people who test positive, what is the probability they actually have the disease Several Events • Suppose (E1, E2, …, Ek) form a partition of the sample space k P( F ) P( F | Ei ) P( Ei ) i 1 • Bayes Theorem: suppose (E1, E2, …, Ek) form a partition of the sample space P( E j | F ) P( F | E j ) P( E j ) k P ( F | E i ) P ( Ei ) i 1 Example (2.28 from text) • • • • Evidence in a paternity suit indicates that 4 particular men are equally likely to be the father of the child Both the child and the mother have type O blood The table below give the blood type and the probability of producing a type O offspring with a type O mother What is the probability that man 1 is the father? Man 3? Man 3? Man 1 2 3 4 Blood Type A B O AB P(O child|man and O-type mother) .431 472 1 0 Independence • For a given probability model and events E and F, the two events are said to be independent iff P(EF)=P(E)P(F) • Another way of viewing this is that for events E and F, the two events are said to be independent iff P(E|F)=P(E) Example • A single card is drawn from a standard deck – – • A={Ace} B={Spade} Are these events independent? Example • Consider a pair of genes which may be one of two alleles a or A – This could represent a smooth (a) or wrinkled seed (A) • According to Medelain Theory, each parent contributes either a or A to the offspring independently • • E1={parent 1 gives a} E2={parent 2 gives a} • P(aa)=