Chapter 11 Probability Sample spaces, events, probabilities, conditional probabilities, independence, Bayes’ formula.

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Chapter 11 Probability Sample spaces, events, probabilities, conditional probabilities, independence, Bayes’ formula

Chapter 12 Sample spaces and events Envision an experiment for which the result is unknown. The collection of all possible outcomes is called the sample space. –Sample spaces can be discrete: {HH, HT, TH, TT} for two coins –Or continuous, e.g., [0,  ), A set of outcomes, or subset of the sample space, is called an event. If E and F are events, –E  F is the event that either E or F (or both) occurs –E  F = EF is the event that both E and F occur. If EF =  then E and F are called mutually exclusive. –The complement of E is : the event that E does not occur

Chapter 13 Probability A probability space is a three-tuple (S, , P) where S is a sample space,  is a collection of events from the sample space and P is a probability law that assigns a number to each event in . P(  ) must satisfy: –P(S) = 1 –0  P(A)  1 –For any collection of mutually exclusive events E 1, E 2, …, –If S is discrete, then  is the set of all subsets of S –If S is continuous, then  can be defined in terms of “basic events of interest,” e.g., if S = [0, 1] the basic events could be all (a, b) with 0  a < b  1. Then  would be the set of intervals along with all their countable unions and intersections.

Chapter 14 Probability If S consists of n equally likely outcomes, then the probability of each is 1/n. Since E and E c are mutually exclusive and E  E c = S, P(E  F) = P(E) + P(F) – P(EF) P E  F  G) = P(E) + P(F) + P(G) – P(EF) – P(EG) – P(FG) + P(EFG) Generalizes to any number of events. Use Venn diagrams!

Chapter 15 Conditional Probabilities If A and B are events with P(B)  0, the conditional probability of A given B is This formula also tells how to find the probability of AB: A and B are independent events if or equivalently if If events are mutually exclusive, are they independent? Vice versa? A set of events E 1, E 2, …, E n are independent if for every subset E 1 ’, E 2 ’, …, E r ’, (pairwise independence is not enough – see Example 1.10)

Chapter 16 Bayes’ Formula The law of total probability says that if E and F are any two events, then since E = EF  EF c, It can be generalized to any partition of S: F 1, F 2, …, F n mutually exclusive events with Bayes’ formula is relevant if we know that E occurred and we want to know which of the F’s occurred.

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