2 n balls are chosen at random from a total of 2 n red balls...

$2 n$ balls are chosen at random from a total of $2 n$ red balls and $2 n$ blue balls. Find a combinatorial expression for the probability that the chosen balls are equally divided in color. Use Stirling's formula to estimate this probability.

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Key Concepts

Asymptotic Estimation

Asymptotic estimation involves approximating mathematical expressions to understand their behavior as variables grow large. This technique is particularly useful when applying Stirling's formula, as it allows one to estimate probabilities in combinatorial contexts without computing large factorials exactly.

Stirling's Formula

Stirling's formula is an asymptotic approximation for factorials, which is essential for estimating expressions involving large factorials. It approximates n! with n^n * e^(-n) * sqrt(2*pi*n) and simplifies calculations in combinatorial probability, especially when n is large.

Combinatorial Probability

This concept involves determining probabilities by counting the number of favorable outcomes divided by the total number of possible outcomes. In combinatorial problems, this is typically done using combinations or binomial coefficients to count selections without regard to order.

Binomial Coefficient

The binomial coefficient, often written as 'n choose k', represents the number of ways to select k items from a set of n items without regard to order. It plays a central role in combinatorial probability as it provides the exact count for various possible outcomes in selection problems.

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