Week3-Class2-conditional

Week3-Class2-conditional - Conditional Probability Enrique...

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Conditional Probability Enrique Campos-N´ a˜nez The George Washington University ApSc 116
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Table of Contents Conditional Probability Event Independence Independence and Conditional Probabilities Bayes’ Theorem
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The Definition of Conditional Probability The quantity Pr ( A | B ) is called the probability of A given B , and it is the probability to be assigned to the event A when it is known that event B has occurred. Definition (Conditional Probability) For events A and B such that Pr ( B ) > 0, we have that Pr ( A | B ) = Pr ( A B ) Pr ( B ) .
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Example (Rolling Dice) Two rice are rolled and the sum T of the two number is odd, what is the probability that T < 8? Let A be the event T < 8, and let B the event T is odd. Then the event AB is the event that T is 3, 5, or 7. Then Pr ( AB ) = 2 + 4 + 6 36 = 12 36 = 1 3 , Pr ( B ) = 2 + 4 + 6 + 4 + 2 36 = 18 36 = 1 2 Therefore, by definition of conditional probability Pr ( A | B ) = Pr ( AB ) Pr ( B ) = 2 3 .
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The Multiplication Rule 1. The multiplication rule. 1.1 From the definition we have Pr ( A B ) = Pr ( B ) Pr ( A | B ) = Pr ( A ) Pr ( B | A ) . 1.2 More generally, Pr ( A B C ) = Pr (( A B ) C ) = Pr ( A B ) Pr ( C | A B ) = Pr ( A ) Pr ( B | A ) Pr ( C | A B ) 1.3 Even more generally, we have Theorem 1: If A 1 , A 2 , . . . , A n are any events such that Pr ( A A 2 ∩ · · · ∩ A n - 1 ) > 0, then Pr ( A 1 A 2 . . . A n ) = Pr ( A 1 ) Pr ( A 2 | A 1 ) Pr ( A 3 | A 1 A 2 ) . . . Pr ( A n | A 1 . . . A n - 1 ) . 2. Exercises: Page 55, #4,6.
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Multiplication Rule Example Example (Selecting Two Balls) Suppose that two balls are to be selected at random, without replacement, from a box containing r red balls and b blue balls. What is the probability that the first ball is red and the second will be blue?
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