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 thenoptions 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. Question: As n→ ∞, determine the exact values (or “limits”) for the probabilities pn(0), pn(1), pn(2), pn(3). Clearly show all of your steps and calculations. Use these results to show that when n is sufficiently large, over 98% of the buckets contain atmost three elements. (This explains why Bucket Sort is such an effective sorting algorithm.)

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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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 thenoptions 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.

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  • As n→ ∞, determine the exact values (or “limits”) for the probabilities pn(0), pn(1), pn(2), pn(3). Clearly show all of your steps and calculations. Use these results to show that when n is sufficiently large, over 98% of the buckets contain atmost three elements. (This explains why Bucket Sort is such an effective sorting algorithm.)
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