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# PROBABILITY BASICS

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### PROBABILITY BASICS

1. 1. Probability is a measure of the likelihood of a random phenomenon or chance behavior. <br />Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty. <br />
2. 2. The Law of Large Numbers<br />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.<br />
3. 3. …Some Definitions…<br />An experiment is any process that can be repeated in which the results are uncertain. <br />A outcome is the result of an experiment. <br />The sample space of a probability experiment is the collection of all possible outcomes. <br />
4. 4. …Some Definitions…<br />An event is a collection of outcomes from a probability experiment. <br />A simple event is one of the possible outcomes…<br />A joint event is more than one of the possible outcomes… <br />
5. 5. …More Definitions…<br />Mutually exclusive events vs. Collectively exhaustive events<br /> <br /> <br /> <br /> <br />at least one of the events must occur<br />cannot occur at the same time (i.e., they have no common outcomes).<br />Independent events vs. Dependent events<br />
6. 6. Three methods for determining the probability of an event: <br />the classical method<br />The classical method of computing probabilities requires equally likely outcomes. <br />An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring.<br />
7. 7. Computing Probability Using the Classical Method<br />If an experiment has equally likely simple events, then the probability of E, P(E), is<br />…need prior information on E…<br />
8. 8. Three methods for determining the probability of an event: <br />(1) the classical method<br />(2) the empirical method<br />(relative frequency)<br />
9. 9. Computing Probability Using the Empirical Method<br />The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the experiment.<br />…need data frequencies from an ‘experiment’ …<br />
10. 10. Three methods for determining the probability of an event: <br />(1) the classical method<br />(2) the empirical method<br />(3) the subjective method<br />
11. 11. Subjective probabilities are probabilities obtained based upon an educated guess. <br />
12. 12. Properties of Probabilities<br />The probability of any event E, P(E), must be between 0 and 1 inclusive. <br />If an event is impossible, the probability of the event is 0.<br />If an event is a certainty, the probability of the event is 1.<br />The sum of the probabilities across all possible outcomes of an experiment is equal to 1.<br /> S = {e1, e2, …, en}, then P(e1) + P(e2) + … + P(en) = 1. <br />
13. 13. …Probability Rules…<br />E<br />
14. 14. …Probability Rules…<br />E<br />
15. 15. E<br />F<br />…Probability Rules…<br />Let E and F be two events… <br />{EandF} is the event consisting of simple events that belong to both E and F. <br />{EorF} is the event consisting of simple events that belong to either E or F but not both. <br />
16. 16. Addition Rule<br />For any two events E and F…<br />P(E or F) = P(E) + P(F) – P(E and F)<br />…Probability Rules…<br />OR probabilities<br />E<br />F<br />
17. 17. …Probability Rules…<br />E<br />F<br />Addition Rule for Mutually Exclusive Events<br />If E and F are mutually exclusive events, then…<br />P(E or F) = P(E) + P(F)<br />cannot occur at the same time (i.e., they have no common outcomes).<br />
18. 18. …Probability Rules…<br />Conditional Probability, P(F|E)<br />The notation P(F | E) is read “the probability of event F given event E”. <br />It is the probability of an event F given the occurrence of the event E. <br />
19. 19. …Probability Rules…<br />
20. 20. …Probability Rules…<br />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. <br />Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F.<br />
21. 21. …Probability Rules…<br />
22. 22. …Probability Rules…<br />