Consider Exercise 30. (a) Calculate the probability of at...

Consider Exercise 30. (a) Calculate the probability of at least one match for $n=2,3,4$, and compare it to $1-e^{-1}$. (b) Show that $$ \left|1-e^{-1}-\left(1-\frac{1}{2 !}+\frac{1}{3 !}-\cdots+\frac{(-1)^{n-1}}{n !}\right)\right| \leq \frac{1}{(n+1) !} . $$ Conclude that, for $n \geq 4$, the probability of at least one match differs from $1-e^{-1} \approx 0.63$ by less than 0.01 ; that is, the probability of no match is about 0.63 for all $n \geq 4$.

Consider Exercise 30.
(a) Calculate the probability of at least one match for $n=2,3,4$, and compare it to $1-e^{-1}$.
(b) Show that
$$
\left|1-e^{-1}-\left(1-\frac{1}{2 !}+\frac{1}{3 !}-\cdots+\frac{(-1)^{n-1}}{n !}\right)\right| \leq \frac{1}{(n+1) !} .
$$
Conclude that, for $n \geq 4$, the probability of at least one match differs from $1-e^{-1} \approx 0.63$ by less than 0.01 ; that is, the probability of no match is about 0.63 for all $n \geq 4$.

Key Concepts

Derangements

Derangements refer to the number of permutations of a set in which none of the elements appear in their original positions. In probability theory and combinatorics, the concept is used to compute the likelihood that a random permutation has no fixed points. This concept is essential in 'matching problems' where an item is not paired with itself.

Exponential Series Approximation

The exponential series, particularly the expansion for the function e^x, is a key concept when approximating probabilities in combinatorial contexts. For example, the series expansion for e^(-1) can be expressed as an alternating series whose partial sums provide approximations to the probability of certain permutation outcomes, particularly in problems involving derangements.

Probability in Combinatorial Settings

In combinatorial probability, counting techniques are used to evaluate the likelihood of various outcomes in scenarios involving permutations and arrangements. Understanding how combinatorial counts, such as those involving factorials and derangements, translate into probabilities is vital for analyzing events where elements may or may not match their prescribed positions.

Alternating Series Error Bound

When approximations are made using alternating series, an important concept is bounding the error of the approximation. The Alternating Series Estimation Theorem provides a means to quantify the maximum deviation between the finite sum and the complete infinite series. This ensures that the approximation is within a specified tolerance, which is particularly useful when comparing calculated probabilities with theoretical constants such as 1 - e^(-1).

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