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lecture6.pdf - COMPSCI 240 Reasoning Under Uncertainty Arya...

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COMPSCI 240: Reasoning Under Uncertainty Arya Mazumdar University of Massachusetts at Amherst Fall 2016
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Lecture 6
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Summary of Counting Problems Structure Description Order Matters Formula Permutation Number of ways to order n ob- jects Yes n ! k- Permutation Number of ways to form a se- quence of size k using k differ- ent objects from a set of n ob- jects Yes n ! ( n - k )! Combination Number of ways to form a set of size k using k different objects from a set of n objects No n ! k !( n - k )! Partition Number of ways to partition n objects into l groups of size k 1 , ..., k l No n ! k 1 ! ... k l !
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 3 aces.
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 3 aces. Answer: There are 4 aces. The probability is ( 4 3 )( 48 4 ) ( 52 7 )
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 2 kings.
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 2 kings. Answer: There are 4 kings. The probability is ( 4 2 )( 48 5 ) ( 52 7 )
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 3 aces or exactly 2 kings, or both.
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 3 aces or exactly 2 kings, or both. Answer: Let A be the event that there are exactly 3 aces; B be the event that there are exactly 2 kings P ( A ) = ( 4 3 )( 48 4 ) ( 52 7 ) ; P ( B ) = ( 4 2 )( 48 5 ) ( 52 7 ) P ( A B ) =?
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 3 aces or exactly 2 kings, or both.
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Example We draw the top 7 cards from a shuffled standard 52-card deck. Find the probability that The 7 cards include exactly 3 aces or exactly 2 kings, or both. Answer: P ( A B ) = ( 4 3 )( 4 2 )( 44 2 ) ( 52 7 ) P ( A B ) = P ( A )+ P ( B ) - P ( A B ) = ( 4 3 )( 48 4 ) ( 52 7 ) + ( 4 2 )( 48 5 ) ( 52 7 ) - ( 4 3 )( 4 2 )( 44 2 ) ( 52 7 )
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The Binomial Law The probability of observing k heads in n independent trials where the probability of success is p in each trial is thus:
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The Binomial Law The probability of observing k heads in n independent trials where the probability of success is p in each trial is thus: P n ( k ) = n k p k (1 - p ) n - k
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Application: Redundancy and Networks I Suppose you’re responsible for the network of web servers that host a large website. If each server has an independent chance of failure of 0 . 1% per day and you have 1000 servers, what is the probability that no servers will fail on any given day?
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