Binomial Distribution
Compute P(X = k) and P(X ≤ k) in binomial distribution B(n, p).
Binomial distribution
The binomial distribution tells you how likely you are to get exactly k successes across n independent Bernoulli trials, each one succeeding with probability p: P(X = k) = C(n,k) · p^k · (1−p)^(n−k), where C(n,k) is the binomial coefficient "n choose k". The mean is μ = n·p and the variance is σ² = n·p·(1−p). Say you flip a fair coin 10 times. The chance of landing exactly 7 heads is C(10,7) · 0.5⁷ · 0.5³ ≈ 0.117, or 11.7%. Once n gets large and both np and n(1−p) clear 5, a Normal(np, np(1−p)) approximates it well, which is the Central Limit Theorem at work.
Applications
Think A/B testing on conversion rates, pass-or-fail counts in exams and selection processes, quality control on defects in a batch, Mendelian inheritance proportions in genetics, opinion polls and the confidence intervals built around proportions, reliability engineering when you want the odds of k failures among n components, and clinical trials measuring response to treatment.
FAQ
When does the binomial model not apply? Two cases break it. One is dependent trials, like sampling without replacement from a small population, where you should switch to the hypergeometric. The other is when p drifts from trial to trial.
Binomial or Poisson? Reach for Poisson when n is very large and p is very small while np stays moderate, the classic rare-event setup. In that regime Poisson tracks the binomial closely.
What's P(X ≤ k)? That is the cumulative distribution, the sum of P(X = i) for i running from 0 to k. You use it to answer "probability of at most k successes".
Related Tools
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.