1. Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. Use the probability applet to simulate flipping a coin 100 times. Plot the proportion of heads against the number of flips. Repeat the simulation. 5-1 © 2010 Pearson Prentice Hall. All rights reserved

2. Probability deals with experiments that yield random short-term results or outcomes, yet reveal long-term predictability. The long-term proportion with which a certain outcome is observed is the probability of that outcome. 5-2 © 2010 Pearson Prentice Hall. All rights reserved

3. The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. 5-3 © 2010 Pearson Prentice Hall. All rights reserved

4. In probability, an experiment is any process that can be repeated in which the results are uncertain. The sample space, S, of a probability experiment is the collection of all possible outcomes. An even is any collection of outcomes from a probability experiment. An event may consist of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events, e i. In general, events are denoted using capital letters such as E. 5-4 © 2010 Pearson Prentice Hall. All rights reserved

5. A probability model lists the possible outcomes of a probability experiment and each outcome’s probability. A probability model must satisfy rules 1 and 2 of the rules of probabilities. 5-5 © 2010 Pearson Prentice Hall. All rights reserved

6. If an event is a certainty, the probability of the event is 1. If an event is impossible, the probability of the event is 0. An unusual event is an event that has a low probability of occurring. 5-6 © 2010 Pearson Prentice Hall. All rights reserved

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8. The classical method of computing probabilities requires equally likely outcomes. An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring. 5-8 © 2010 Pearson Prentice Hall. All rights reserved

9. The subjective probability of an outcome is a probability obtained on the basis of personal judgment. For example, an economist predicting there is a 20% chance of recession next year would be a subjective probability. 5-9 © 2010 Pearson Prentice Hall. All rights reserved

10. Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. 5-10 © 2010 Pearson Prentice Hall. All rights reserved

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13. Complement of an Event Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted E C, is all outcomes in the sample space S that are not outcomes in the event E. 5-13 © 2010 Pearson Prentice Hall. All rights reserved

14. Complement Rule If E represents any event and E C represents the complement of E, then P(E C ) = 1 – P(E) 5-14 © 2010 Pearson Prentice Hall. All rights reserved

15. Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F. 5-15 © 2010 Pearson Prentice Hall. All rights reserved

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19. Conditional Probability The notation P(F | E) is read “the probability of event F given event E”. It is the probability of an event F given the occurrence of the event E. 5-19 © 2010 Pearson Prentice Hall. All rights reserved

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22. A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol n C r represents the number of combinations of n distinct objects taken r at a time, where r < n. 5-22 © 2010 Pearson Prentice Hall. All rights reserved

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25. ●A partition of the sample space S are two non-empty sets A 1 and A 2 that divide up S ●In other words  A 1 ≠ Ø  A 2 ≠ Ø  A 1 ∩ A 2 = Ø (there is no overlap)  A 1 U A 2 = S (they cover all of S) 5-25 © 2010 Pearson Prentice Hall. All rights reserved

26. ●Let E be any event in the sample space S ●Because A 1 and A 2 are disjoint, E ∩ A 1 and E ∩ A 2 are also disjoint ●Because A 1 and A 2 cover all of S, E ∩ A 1 and E ∩ A 2 cover all of E ●This means that we have divided E into two disjoint pieces E = (E ∩ A 1 ) U (E ∩ A 2 ) 5-26 © 2010 Pearson Prentice Hall. All rights reserved

27. ●Because E ∩ A 1 and E ∩ A 2 are disjoint, we can use the Addition Rule P(E) = P(E ∩ A 1 ) + P(E ∩ A 2 ) ●We now use the General Multiplication Rule on each of the P(E ∩ A 1 ) and P(E ∩ A 2 ) terms P(E) = P(A 1 ) P(E | A 1 ) + P(A 2 ) P(E | A 2 ) 5-27 © 2010 Pearson Prentice Hall. All rights reserved

28. P(E) = P(A 1 ) P(E | A 1 ) + P(A 2 ) P(E | A 2 ) ●This is the Rule of Total Probability (for a partition into two sets A 1 and A 2 ) ●It is useful when we want to compute a probability (P(E)) but we know only pieces of it (such as P(E | A 1 )) ●The Rule of Total Probability tells us how to put the probabilities together 5-28 © 2010 Pearson Prentice Hall. All rights reserved

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30. ●The general Rule of Total Probability assumes that we have a partition (the general definition) of S into n different subsets A 1, A 2, …, A n  Each subset is non-empty  None of the subsets overlap  S is covered completely by the union of the subsets ●This is like the partition before, just that S is broken up into many pieces, instead of just two pieces 5-30 © 2010 Pearson Prentice Hall. All rights reserved

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32. The Rule of Total Probability – Know P(A i ) and P(E | A i ) – Solve for P(E) Bayes’s Rule – Know P(E) and P(E | A i ) – Solve for P(A i | E) 5-32 © 2010 Pearson Prentice Hall. All rights reserved

33. Bayes’ Rule, for a partition into two sets U 1 and U 2, is This rule is very useful when P(U 1 |B) is difficult to compute, but P(B|U 1 ) is easier 5-33 © 2010 Pearson Prentice Hall. All rights reserved

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