Calculate binomial probabilities using the coin flip model.
Last updated: March 2026
| n Flips | k Heads | P(X = k) | P(X ≥ k) | Expected Value |
|---|---|---|---|---|
| 5 | 2 | 31.25% | 50% | 2.5 |
| 10 | 5 | 24.61% | 62.30% | 5 |
| 10 | 6 | 20.51% | 37.70% | 5 |
| 20 | 10 | 17.62% | 58.81% | 10 |
Binomial probability calculates the likelihood of obtaining exactly k successes in n independent trials, where each trial has the same probability p of success. It's one of the most important probability distributions in statistics.
The classic example is a coin flip: if you flip a fair coin 10 times, what's the probability of getting exactly 6 heads? The binomial distribution answers this question. The formula is: P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where C(n,k) is the binomial coefficient.
Binomial probability applies to any scenario with two outcomes (success/failure, heads/tails, yes/no) where trials are independent and the probability of success is constant. It's widely used in quality control, medical testing, polling, and risk analysis.
Flip a fair coin 10 times. What's P(exactly 6 heads)?
For a truly fair coin, p = 0.5. Unfair coins (p ≠ 0.5) will shift the probabilities. The calculator works with any p value for weighted or biased scenarios.
P(X = k) is the exact probability of exactly k successes. P(X ≥ k) is the cumulative probability of k or more successes. Use ≥ for 'at least' questions.
They measure the spread of outcomes. High variance means results are unpredictable; low variance means results tend to cluster around the expected value.
Yes! Any two-outcome scenario works: passing/failing a test, converting/not converting a customer, etc. Just adjust p to match your success rate.
The binomial distribution approximates the normal distribution for large n. Use a normal distribution calculator for very large n (n > 30) for faster computation.
Expected value = n × p is the average number of successes if you repeated the experiment many times. For 10 flips with p=0.5, you expect 5 heads on average.
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