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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2009 • Question 5

Question

Math 6010 • Linear Models Fall 2009 • Question 5

Consider a deck of \(n\) cards labeled \(1,2,\ldots ,n\). Assume it is well shuffled with every possible arrangement equally likely. Let \(E_n\) be the event that card \(j\) is in position \(j\) in the shuffled deck for some \(j\in \{1,2,\ldots ,n\}\). Evaluate \(\mathbb {P}(E_n)\) using inclusion-exclusion and show that \(\lim _{n\to \infty }\mathbb {P}(E_n)=1-e^{-1}\).

Raw LaTeX

Consider a deck of \(n\) cards labeled \(1,2,\ldots,n\). Assume it is
well shuffled with every possible arrangement equally likely. Let \(E_n\) be the
event that card \(j\) is in position \(j\) in the shuffled deck for some
\(j\in\{1,2,\ldots,n\}\). Evaluate \(\Prob(E_n)\) using inclusion-exclusion and
show that \(\lim_{n\to\infty}\Prob(E_n)=1-e^{-1}\).

Topic Tags

Classical Probability and Inclusion-Exclusion Uniform Integrability and Truncation Methods