Hypergeometric Probabilities

What this lecture covers

This short problem derives the hypergeometric probability formula: given an urn of n balls where m are red, what is the probability that exactly i of k balls drawn (without replacement) are red? It applies the discrete uniform law together with counting arguments already developed in the course, then works a numerical example using a deck of cards.

After watching, you should be able to derive and apply the hypergeometric probability formula, and recognize problems that fit this pattern, such as drawing a fixed number of a specific card rank from a deck.

Key ideas

• Hypergeometric setup: an urn has n balls total, m of which are red; k balls are drawn without replacement, and the question asks for the probability that exactly i are red.
• Sample space size: the total number of ways to draw k balls out of n is n choose k.
• Counting favorable outcomes: the number of ways to get exactly i red balls is the product of choosing i red balls from the m red balls and choosing the remaining k-i balls from the n-m non-red balls, i.e. (m choose i) * (n-m choose k-i).
• Hypergeometric formula: combining these gives P = (m choose i)(n-m choose k-i) / (n choose k).
• Card example: for a 52-card deck with 4 aces, drawing 7 cards, the probability of getting exactly 3 aces is (4 choose 3)(48 choose 4) / (52 choose 7).

Before you watch

• Review the discrete uniform law and binomial coefficients (n choose k) from earlier lectures in this course.

Check your understanding

1. Why is the total number of ways to draw k balls from n equal to n choose k?
2. Why does counting the favorable outcomes require multiplying two separate binomial coefficients rather than adding them?
3. How would the formula change if the balls were drawn with replacement instead?