Binomial Probability Calculator
The chance of exactly, at least or at most k successes in n tries, with the mean, standard deviation and the full distribution.
For example, coin tosses, questions guessed or items checked.
P(X ≥ 7)
0.171875
- As a percentage
- 17.1875%
- P(X = 7)
- 0.117188
- Expected successes (mean)
- 5
- Standard deviation
- 1.5811
With 10 tries at a 50% chance each, the chance of at least 7 successes is 17.19%, about 1 in 5.8.
How it's worked out
- Chance of exactly j successes
C(n, j) × p^j × (1 − p)^(n − j)binomial formula - Exactly 7
C(10, 7) × 0.5^7 × 0.5^30.117188 - Add up every j with j ≥ 7
j from 7 to 100.171875 - Mean and standard deviation
n × p and √(n × p × (1 − p))5 and 1.5811
Chance of each number of successes
| Successes | Probability | At most this many |
|---|---|---|
| 0 | 0.000977 | 0.000977 |
| 1 | 0.009766 | 0.010742 |
| 2 | 0.043945 | 0.054688 |
| 3 | 0.117188 | 0.171875 |
| 4 | 0.205078 | 0.376953 |
| 5 | 0.246094 | 0.623047 |
| 6 | 0.205078 | 0.828125 |
| 7 | 0.117188 | 0.945313 |
| 8 | 0.043945 | 0.989258 |
| 9 | 0.009766 | 0.999023 |
| 10 | 0.000977 | 1 |
- The binomial model needs a fixed number of independent tries, each with the same chance of success.
The formula
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)
For “at least”, “at most”, “fewer than” and “more than”, the calculator adds up P(X = j) for every j in the range. It works in logarithms, so even thousands of tries don't overflow.
Mean and spread
Mean = n × p · Standard deviation = √(n × p × (1 − p))
With 10 fair coin tosses you expect 5 heads, give or take 1.58.
Questions people ask
What is the chance of at least 7 heads in 10 coin tosses?
176 ÷ 1,024 = 0.171875, about 17.2%, or 1 in 5.8.
What is the chance of rolling at least one six in six rolls?
1 − (5/6)⁶ = 0.6651, about 66.5%. It's easier to work out the chance of no sixes and take it from 1.
What is the binomial formula?
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where C(n, k) is the number of ways to choose k successes from n tries.
When can I use the binomial distribution?
When there's a fixed number of tries, each try is independent, and each has the same chance of success, such as coin tosses or guessing answers at random.
About this calculator
Formula and rules last checked 9 November 2026Written and tested by the CalculatorDex teamReport a problem