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MAT 119 FALL 2001 MAT 119 FINITE MATHEMATICS NOTES PART 2 – PROBABILITY CHAPTER 7 PROBABILITY 7.1 Sample Spaces and the Assignment of Probabilities Sample space, S – includes all outcomes (element) that can occur in a real or conceptual experiment Eg; The experiment of tossing a coin once S = {H, T} The experiment of tossing a coin once S = {HH, HT, TH, TT} Constructing a Probability Model 1. Find the sample space. List all outcomes, if too many, find the number of outcomes. 2. Assign each e a probability P(e) so that a. b. P(e) 0 sum of probabilities assigned to all outcomes is 1. DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 1 MAT 119 FALL 2001 Event – any subset of the sample space (2 or more outcomes) simple event – has only one outcome Probability of an Event If E = Ø, the event is impossible and P(Ø) = 0 If E = {e} is a simple event, P(E) = P(e) is the probability assigned to outcome e. If E is the union of r simple events {e1}, {e2}, …, {er} and P(E) = P(e1) + P(e2) + … + P(er) DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 2 MAT 119 FALL 2001 7.2 Properties of the Probability of an Event Mutually exclusive events (2 or more from s sample space) – have no common outcomes Let E and F be mutually exclusive events, EF= P(E or F) = P(E F) = P(E) + P(F) since P(E F) = 0 Additive Rule For any two events A and B of a sample space S P(E F) = P(E) + P(F) – P(E F) Properties of an Event 1. 0 P(E) 1 2. P() = 0 3. P(E F) = P(E) + P(F) – P(E F) and P(S) = 1 Complement E of E, P( E ) = 1 – P(E) DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 3 MAT 119 FALL 2001 Let a sample space S be given by S = {e1, e2, …, en} where all outcomes are equally likely. For an event E containing m outcomes, P(E) = m/n Probability of an event E in a sample space with equally likely outcomes If the sample space S of an experiment has n equally likely outcomes, and the event E in S occurs m times, then P( E ) Number of possible ways the event E can take place c( E ) Number of outcomes in S c( S ) If the odds for E are a to b, then P( E ) a ab If the odds against E are a to b, then P( E ) b ab DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 4 MAT 119 FALL 2001 7.3 Probability problems using counting techniques. DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 5 MAT 119 FALL 2001 7.4 Conditional Probability Let E and F be events of a sample space S and suppose P(F) > 0. The conditional probability of the event E, assuming the event F, denoted by P(E/F), is defined as P( E / F ) P( E F ) P( F ) Product Rule - P(E F) = P(E) . P(E/F) DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 6 MAT 119 FALL 2001 7.5 Independent Events E is independent of F, if and only if, P(E/F) = P(E) Thm. Let E and F be events where P(E) > 0 and P(F) > 0. If E is independent of F, then F is independent of E. If two events E and F have positive probabilities and if event E is independent of F, then F is also independent of E. E and F are called independent events. Test for Independence Two events E and F of a sample space S are independent events if and only if P(E F) = P(E) . P(F) In words, the probability of E and F is equal to the product of the probability of E and the probability of F. A set E1, E2, …, En of n events is called independent if the occurrence of one or more of them does not change the probability of any of the others. It can be shown that, for such events, P(E1 E2 … En) = P(E1) . P(E2) . … . P(En) DOUGLAS A. WILLIAMS, ARIZONA STATE UNIVERSITY, DEPARTMENT OF MATHEMATICS 7

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