P E F P EF P F if P F 0 In example with dice P E F 1 36 1 6 1 6 Conditional

# P e f p ef p f if p f 0 in example with dice p e f 1

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P ( E | F ) = P ( EF ) P ( F ) , if P ( F ) > 0. In example with dice P ( E | F ) = 1 / 36 1 / 6 = 1 6 . Conditional Probability EXAMPLE 2b: A coin is flipped twice. What is the conditional probability that both flips land on heads, given that at least one flip lands on heads? Conditional Probability EXAMPLE 2b: A coin is flipped twice. What is the conditional probability that both flips land on heads, given that at least one flip lands on heads? We are assuming that all four points in the sample space S = { ( h , h ) , ( h , t ) , ( t , h ) , ( t , t ) } are equally likely. Let B = { ( h , h ) } be the event that both flips land on heads, and let A = { ( h , h ) , ( h , t ) , ( t , h ) } be the event that at least one flip lands on heads. Conditional Probability EXAMPLE 2b: A coin is flipped twice. What is the conditional probability that both flips land on heads, given that at least one flip lands on heads? We are assuming that all four points in the sample space S = { ( h , h ) , ( h , t ) , ( t , h ) , ( t , t ) } are equally likely. Let B = { ( h , h ) } be the event that both flips land on heads, and let A = { ( h , h ) , ( h , t ) , ( t , h ) } be the event that at least one flip lands on heads. P ( B | A ) = P ( BA ) P ( A ) = P ( { ( h , h ) } ) P ( { ( h , h ) , ( h , t ) , ( t , h ) } ) = 1 / 4 3 / 4 = 1 3 Conditional Probability EXAMPLE 2a: A student is taking a one-hour-time-limit makeup examination. Sup- pose the probability that the student will finish the exam in less than x hours is x / 2, for all 0 x 1. Then, given that the student is still working after . 75 hour, what is the conditional probability that the full hour is used? Conditional Probability EXAMPLE 2a: A student is taking a one-hour-time-limit makeup examination. Sup- pose the probability that the student will finish the exam in less than x hours is x / 2, for all 0 x 1. Then, given that the student is still working after . 75 hour, what is the conditional probability that the full hour is used? Let L x denote the event that the student finishes the exam in less than x hours, 0 x 1, and let E be the event that the student uses the full hour. We need to find P ( E | L c . 75 ). Conditional Probability EXAMPLE 2a: A student is taking a one-hour-time-limit makeup examination. Sup- pose the probability that the student will finish the exam in less than x hours is x / 2, for all 0 x 1. Then, given that the student is still working after . 75 hour, what is the conditional probability that the full hour is used? Let L x denote the event that the student finishes the exam in less than x hours, 0 x 1, and let E be the event that the student uses the full hour. We need to find P ( E | L c . 75 ). P ( E | L c . 75 ) = P ( EL c . 75 ) P ( L c . 75 ) = P ( E ) 1 - P ( L . 75 ) = . 5 1 - . 75 / 2 = . 8 Conditional Probability EXAMPLE 2c: In the card game bridge, the 52 cards are dealt out equally to 4 players - called East, West, North, and South. If North and South have a total of 8 spades among them, what is the probability that East has exactly 3 of the remaining 5 spades? Conditional Probability EXAMPLE 2c: In the card game bridge, the 52 cards are dealt out equally to 4 players - called East, West, North, and South. If North and South have a total of 8 spades among them, what is the probability that East has exactly 3 of the remaining 5 spades?  #### You've reached the end of your free preview.

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