Hypergeometric Distribution Calculator
Statistics, Probability
Intro: Computes hypergeometric probabilities for draws without replacement from a finite population.
Worked example
- A lot has N=50 items, K=8 are defective. If n=5 are sampled without replacement, find P(X=2 defective).
- For a hypergeometric distribution, $P(X=k)=\dfrac{{K \choose k}{N-K \choose n-k}}{{N \choose n}}$.
- Here $N=50$ (total items), $K=8$ (defective), $n=5$ (draws), and $k=2$ (number of defectives we want).
- Compute the numerator combination for successes: ${K \choose k} = {8 \choose 2} = \dfrac{8\cdot7}{2\cdot1} = 28$.
- Compute the numerator combination for non-successes: $N-K = 50-8=42$, so ${N-K \choose n-k} = {42 \choose 3} = \dfrac{42\cdot41\cdot40}{3\cdot2\cdot1} = 11480$.
- Multiply these to get the numerator: $28 \times 11480 = 321,440$.
- Compute the denominator: ${N \choose n} = {50 \choose 5} = \dfrac{50\cdot49\cdot48\cdot47\cdot46}{5\cdot4\cdot3\cdot2\cdot1} = 2,118,760$.
- So $P(X=2)=\dfrac{321,440}{2,118,760} \approx 0.152$.
- Answer: $\boxed{P(X=2) \approx 0.152}$, about a $15.2\%$ chance of getting exactly 2 defective items.
FAQs
When do I use hypergeometric instead of binomial?
Use hypergeometric when sampling is without replacement from a finite population; use binomial when each trial is independent with the same probability p.
Why choose Mathix?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.