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Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of the n options being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul- tiple objects. For each integer k with 0 < k < n, define pn (k) to be the probability that a randomly-chosen bucket contains exactly k objects. By definition, pn(0) + pn (1) + Pn(2) +. . .+ Pn (n – 1) + Pn(n) = 1. (a) Determine each of the following probabilities: p4(0), p4(1), p4(2), p4(3), p4(4). Briefly explain how you computed each of these probabilities.

Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of the n options being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul- tiple objects. For each integer k with 0 < k < n, define pn (k) to be the probability that a randomly-chosen bucket contains exactly k objects. By definition, pn(0) + pn (1) + Pn(2) +. . .+ Pn (n – 1) + Pn(n) = 1. (a) Determine each of the following probabilities: p4(0), p4(1), p4(2), p4(3), p4(4). Briefly explain how you computed each of these probabilities.

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.2: Probability
Problem 2E
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Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of the n options being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul- tiple objects. For each integer k with 0 < k < n, define pn (k) to be the probability that a randomly-chosen bucket contains exactly k objects. By definition, pn(0) + pn (1) + Pn(2) +. . .+ Pn (n – 1) + Pn(n) = 1. (a) Determine each of the following probabilities: p4(0), p4(1), p4(2), p4(3), p4(4). Briefly explain how you computed each of these probabilities. () - (b) In class, we showed that pPn(k) 1 1- - n-k where k n! k!(n – k)!" n Clearly explain why this formula is true for all integers n and k with 0 < k < n.
Transcribed Image Text:Let n be a positive integer. Suppose you have n objects and n buckets. For each object, randomly place it into a bucket, with each of the n options being equally likely. Through this random process, some buckets may end up empty, while other buckets may have mul- tiple objects. For each integer k with 0 < k < n, define pn (k) to be the probability that a randomly-chosen bucket contains exactly k objects. By definition, pn(0) + pn (1) + Pn(2) +. . .+ Pn (n – 1) + Pn(n) = 1. (a) Determine each of the following probabilities: p4(0), p4(1), p4(2), p4(3), p4(4). Briefly explain how you computed each of these probabilities. () - (b) In class, we showed that pPn(k) 1 1- - n-k where k n! k!(n – k)!" n Clearly explain why this formula is true for all integers n and k with 0 < k < n.
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