Q.Two numbers are selected at random (without replacement) from the first five positive integers. Let $X$ denote the larger of the two numbers obtained. Find the mean and variance of $X$.

Hypergeometric Probability: Drawing Without Replacement

You have a bag of 20 marbles: 12 red and 8 blue. You pick 5 without putting any back. What's the chance exactly 3 are red?

This is what the hypergeometric distribution handles. Unlike the binomial distribution, the trials are not independent — each draw changes the composition of the bag.

The Intuition

When you draw without replacement, the probability of a red on the second draw depends on the first: take a red first and fewer reds remain, so the next red is less likely. Hypergeometric probability captures exactly this dependency.

The setup: "I have a finite population split into two groups. I take a sample without replacement. What's the probability my sample has exactly $k$ items from the first group?"

The Precise Statement

Where:

• $N$ = total items in the population (20 marbles)
• $K$ = number of "success" items ($12$ red)
• $n$ = number drawn (sample size, $5$)
• $k$ = successes wanted in the sample ($3$ reds)

Why This Formula Makes Sense

The denominator $\binom{N}{n}$ counts all ways to choose $n$ items from $N$ — the equally likely outcomes. The numerator counts favourable ones:

• $\binom{K}{k}$: choose $k$ reds from the $K$ reds
• $\binom{N-K}{n-k}$: choose the remaining $n-k$ from the $N-K$ blues

Multiplying pairs each way of picking reds with each way of picking blues.

Worked Example

$N=20$, $K=12$, $n=5$, $k=3$:

About 39.7%.

When to Use Hypergeometric …