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The Probability that a Queen Sits Beside a King: Juxtapositions and Runs in a Random Permutation (UMAP)

John M. Holte, Mark M. Holte, and Kenneth A. Suman

 Mathematics Topic:Probability, Statistics Application Areas:Statistics, applied probability, recreational mathematics Prerequisites:Combinatorics (addition/multiplication principles, permutations, combinations, and distributions of balls into urns); probability; statistics

| ©1999 by COMAP, Inc. | Tools for Teaching 1998 | 22 pages |

This module deals with random permutations, focusing on probabilities, means, and variances of the numbers of juxtapositions and runs of items of different types. It illustrates the ideas via questions regarding a shuffled deck of standard playing cards, such as: What is the probability that a Queen is next to a King? What is the average number of juxtapositions? What is the variance? Tough problems are tamed. In a statistics course, this module provides a natural extension of the usual distribution theory of runs.

INTRODUCTION

THE PROBABILITY THAT A QUEEN IS NEXT TO A KING

JUXTAPOSITIONS AND RUNS

ORDERED JUXTAPOSITIONS

MEAN AND VARIANCE

RELATED PROBLEMS

RANDOM CIRCULAR PERMUTATIONS

SOLUTIONS

REFERENCES