An urn contatns 2 n balls, of which 2 are numbered 1,2 are numbered 2, …and 2 are numbered n. Ba

step 1 Dear students, a solution for problem 22 of chapter 4, section 2, solution. Solution for part fi

An urn contatns 2 n balls, of which 2 are numbered 1,2 are...

An urn contatns $2 n$ balls, of which 2 are numbered 1,2 are numbered $2, \ldots$ and 2 are numbered $n$. Balls are successively withdrawn 2 at a time without replacement. Let $T$ denote the first selection in which the balls withdrawn have the same number (and let it equal infinity if none of the pairs withdrawn has the same number). For $0<\alpha<1$ we want to show that $$ \lim _{n} P\{T>\alpha n\}=e^{-\alpha / 2} $$ To verify the above, let $M_{k}$ denote the number of pairs withdrawn in the first $k$ selections, $k=1, \ldots, n$ (a) Argue that when $n$ is large, $M_{k}$ can be regarded as the number of successes in $k$ (approximately) independent trials. (b) When $n$ is large, approximate $P\left\{M_{k}=0\right\}$. (c) Write the event $\{T>\alpha n\}$ in terms of the value of one of the variables $M_{k}$. (d) Verify the limiting probability above.

An urn contatns $2 n$ balls, of which 2 are numbered 1,2 are numbered $2, \ldots$ and 2 are numbered $n$. Balls are successively withdrawn 2 at a time without replacement. Let $T$ denote the first selection in which the balls withdrawn have the same number (and let it equal infinity if none of the pairs withdrawn has the same number). For $0<\alpha<1$ we want to show that
$$
\lim _{n} P\{T>\alpha n\}=e^{-\alpha / 2}
$$
To verify the above, let $M_{k}$ denote the number of pairs withdrawn in the first $k$ selections, $k=1, \ldots, n$
(a) Argue that when $n$ is large, $M_{k}$ can be regarded as the number of successes in $k$ (approximately) independent trials.
(b) When $n$ is large, approximate $P\left\{M_{k}=0\right\}$.
(c) Write the event $\{T>\alpha n\}$ in terms of the value of one of the variables $M_{k}$.
(d) Verify the limiting probability above.

Key Concepts

Sampling Without Replacement

This concept refers to the process of drawing items from a set (here, balls from an urn) such that once an item is drawn, it is not returned to the set. In these kinds of problems, this leads to dependency between successive draws. However, when the total number of items is large compared to the number drawn, the effect of such dependency can be minimized, which allows for useful approximations.

Approximation of Nearly Independent Trials

In large systems, events that are inherently dependent can, under appropriate conditions, be approximated as independent. In this problem, the successive trials of drawing pairs of balls are approximately independent when n (the number of pairs available) is large. This approximation simplifies the analysis because it allows us to treat the occurrence of matching pairs as roughly Bernoulli trials, each with its own success probability.

Binomial Distribution Approximation

When a series of independent (or nearly independent) Bernoulli trials is conducted, the number of successes in those trials follows a binomial distribution. In the context of this problem, the number M?, which counts the number of matching pairs in k selections, can be approximated by a binomial random variable. This approximation is key to analyzing the probability of having no matching pairs in the first k trials.

Asymptotic Analysis

Asymptotic analysis is the study of the behavior of functions or processes as some parameter (here, n) grows without bound. In this problem, by considering the limit as n becomes large, we can simplify the calculations and derive an elegant limiting probability expression. This technique is central to understanding many probabilistic systems in which exact computation is intractable for finite n.

Exponential Limit in Probability

The exponential limit emerges often in probability theory when dealing with rare events or as a result of limit theorems. In this problem, by estimating the probability of no success (i.e., no matching pair in the first ?n selections) using the binomial approximation, and then taking the limit as n tends to infinity, we arrive at the expression e^(??/2). This step illustrates how combinatorial probabilities can converge to an exponential form when appropriately normalized.

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