Coin Flip Probability Calculator
Enter the number of flips, the probability of heads, and the number of heads you care about to compute exact binomial probabilities.
Exact Probability P(X = k)
24.61%
Exact number of heads
Cumulative P(X ≤ k)
62.30%
At most k heads
Cumulative P(X ≥ k)
62.30%
At least k heads
| Heads | Probability |
|---|---|
| 0 | 0.10% |
| 1 | 0.98% |
| 2 | 4.39% |
| 3 | 11.72% |
| 4 | 20.51% |
| 5 | 24.61% |
| 6 | 20.51% |
| 7 | 11.72% |
| 8 | 4.39% |
| 9 | 0.98% |
| 10 | 0.10% |
How to Use This Calculator
- Enter the number of coin flips (trials) and the probability of heads.
- Specify how many heads you are interested in.
- Read off the exact probability of seeing that many heads along with cumulative probabilities.
- Use the distribution table to understand the entire binomial probability mass function.
Formula
- C(n, k) is the binomial coefficient n choose k.
- p is the probability of heads on a single flip.
- The distribution is valid for any Bernoulli process with independent trials.
Full Description
The binomial distribution models the number of successes in n independent trials. For coin flips, “success” typically represents drawing heads. This calculator computes exact probabilities by evaluating the binomial formula, avoiding approximations. It also provides cumulative probabilities, which are useful for hypothesis testing and risk assessments.
Understanding the distribution helps quantify how surprising unusual sequences are. For example, seeing eight or more heads in ten flips of a fair coin occurs less than 5% of the time — a fact you can verify instantly with this tool.
Frequently Asked Questions
When should I use a normal approximation?
For large n, the binomial distribution becomes expensive to compute manually. A normal approximation works well when np and n(1 − p) are both large, but this calculator provides exact values up to 200 trials.
Can I calculate cumulative probabilities for ranges?
Yes. Use the distribution table to sum any range of outcomes manually, or leverage the P(X ≤ k) and P(X ≥ k) summaries for quick bounds.
Does p have to equal 0.5?
No. Set p to any value between 0 and 1 to model biased coins or weighted Bernoulli trials.
How precise are the probabilities?
Calculations use double-precision arithmetic. For very large n, rounding errors may appear, but within the supported range the accuracy is excellent for practical purposes.