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Mt139C7 - Probability Rules The Probability P(A of any...

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MT139~7.PPT 1 Probability Rules The Probability P(A) of any Event A is between 0 and 1 If S is the Sample Space, then P(S) = 1 For any Event A, the Probability that A does NOT occur is 1 - P(A) The Addition Rule for Disjoint Events says that if A and B are disjoint, P(A or B) = P(A) + P(B). Example: Tossing a coin 3 times. What is the probability of getting two heads and a tail? Probability Model for Three Tosses of a Fair Coin S = HHH HHT HTH THH HTT TTH THT TTT P = 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 Event A = (HHT, HTH, THH) HHT, HTH, THH are Disjoint so P(A) = 1/8 + 1/8 + 1/8 = 0.375 Probability of not getting exactly 2 heads is 1 - 0.375 = 0.625
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MT139~7.PPT 2 Venn Diagram A Venn Diagram Shows the sample space as a rectangular region with Circles representing the probability of one or more events. Venn Diagrams communicate whether two events are disjoint. If two events are not disjoint, the regions of probability will overlap showing the probability that both events would occur. Example: Two tosses of a fair coin. Event A is heads on the first toss and Event B is heads on the second toss. .25 .25 .25 .25 T H Toss 1 T H Toss 2 .50 .50 .50 .50 A B A & B Neither A Nor B
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MT139~7.PPT 3 Independence Two events A and B are Independent if knowing that one occurs does not change the probability that the other occurs. Example: The outcome of a single draw of the Lottery does not change the probability that the same outcome would occur the very next night. The individual Lottery drawings are Independent. “Independent” is not the same as “disjoint”. In fact, if two events are disjoint, they cannot be independent. If A and B are disjoint, knowing that an event A occurred changes the probability of event B to zero.
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MT139~7.PPT 4 Multiplication Rule for Independent Events If A and B are independent events, Example: Getting heads on the second toss of a coin is Independent of whether a heads was tossed on the first. The probability of heads on a single toss of a coin is 0.50. The probability of heads on both tosses of a coin is: Example: A machine will fail if either of its two parts fail. Failure of one part is independent of the other. Each has a probability of failure of 0.03. The probability that the machine will function is: ) B ( P ) A ( P ) B and A ( P = 25 . 0 ) 50 . 0 ( ) 50 . 0 ( ) H ( P ) H ( P ) H and H ( P = = = 941 . 0 ) 97 . 0 97 . 0 ( ) 03 . 0 1 ( ) 03 . 0 1 ( ) Fail ( P 1 [ )] Fail ( P 1 [ ) Function Both ( P = = - - = - - =
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MT139~7.PPT 5 The General Addition Rule For any two events A and B, Example: On any given day, the probability that I will go out to lunch (A) is 0.10. The probability that I will wash my car (B) over my lunch hour is 0.05. The probability that I will do both (A and B) on the same lunch hour is 0.02. On a given day, the probability that I will either go out to lunch or get my car washed is: ) B and A ( P ) B ( P ) A ( P ) B or A ( P - + = 13 . 0 02 . 0 05 . 0 10 . 0 ) B or A ( P = - + =
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MT139~7.PPT 6 Conditional Probability Conditional probability is the probability of one event given that one or more other events are already known to have occurred.
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