Grinstead-Snell-Probability · Problem 33

The core concept is hypergeometric probability: you're sampling without replacement from a finite population, and you want the probability of getting exactly $n$ red and $n$ blue balls. The key insight is that the number of favorable outcomes (ways to choose $n$ red from $2n$ red and $n$ blue from $2n$ blue) divided by total outcomes (ways to choose $2n$ balls from $4n$ total) gives you a ratio of binomial coefficients. Stirling's formula then lets you approximate this ratio for large $n$ by converting factorials into simpler asymptotic forms, revealing how the probability scales with $n$.

The core concept is hypergeometric probability: you're sampling without replacement from a finite population, and you want the probability of getting exactly $n$ red and $n$ blue balls. The key insight is that the number of favorable outcomes (ways to choose $n$ red from $2n$ red and $n$ blue from $2n$ blue) divided by total outcomes (ways to choose $2n$ balls from $4n$ total) gives you a ratio of binomial coefficients. Stirling's formula then lets you approximate this ratio for large $n$ by converting factorials into simpler asymptotic forms, revealing how the probability scales with $n$.