Frequently Asked Questions
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the probability of exactly k successes. P(X ≤ k) is the cumulative probability of k or fewer successes - the sum of P(X = 0) through P(X = k). For the 10 flip / 5 heads example, P(X = 5) ≈ 0.246 while P(X ≤ 5) ≈ 0.623.
Why is the most likely outcome not very probable?
In a binomial distribution, probability is spread across n + 1 possible counts. Even the mode (most likely value) may have only 20-25% probability when n is moderate. The full distribution, not just the mode, carries the information.
Can I use p = 0 or p = 1?
Yes. With p = 0 every trial fails, so P(X = 0) = 1 and all other probabilities are 0. With p = 1 every trial succeeds, so P(X = n) = 1. These are degenerate cases but are handled correctly.
When should I use a Poisson approximation instead?
The Poisson approximation is good when n is large (n > 100) and p is small (p < 0.01), making np the rate λ. For moderate n and p not close to 0 or 1, the exact binomial is the right tool.
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This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.