Probability of Success Calculator
Counting Outcomes Across Repeated Trials
Flip a biased coin ten times, screen a batch of parts for defects, or track click-throughs on an ad campaign, and the same underlying model applies: a fixed number of independent trials, each with the same probability of success. The binomial distribution describes exactly how likely each possible outcome count is, and this calculator evaluates it in four different ways depending on the question being asked.
The Formula
where n = number of trials, k = number of successes, p = probability of success per trial, C(n, k) = n! / (k!(n−k)!)
The "at least," "at most," and "between" modes sum this expression across the relevant range of k values.
Worked Example: 10 Trials, p = 0.5
The full probability distribution for 10 coin flips, computed directly from the formula above:
| Successes (k) | P(X = k) |
|---|---|
| 0 | 0.098% |
| 2 | 4.395% |
| 4 | 20.508% |
| 5 | 24.609% |
| 6 | 20.508% |
| 8 | 4.395% |
| 10 | 0.098% |
The full distribution is symmetric around k = 5 because p = 0.5; it would skew for any other probability.
Where This Calculation Matters
- Quality control — estimating the probability of finding a given number of defective units in a sampled batch, given a known defect rate.
- Marketing and conversion analysis — modeling how many conversions to expect from a fixed number of visitors at a known conversion rate, or the probability of exceeding a target.
- Games of chance and risk modeling — computing the odds of a specific run of outcomes across repeated independent events, from dice games to reliability testing.
How to Use This Calculator
- Choose a calculation type: Exactly K Successes, At Least K Successes, At Most K Successes, or Between K1 and K2 Successes.
- Enter the number of trials (n).
- Enter the probability of success per trial (a value between 0 and 1).
- Enter the relevant success count(s) — k, or k1 and k2 for the range mode.
- Select Calculate to get the probability as a percentage.
Related Calculations
For the general shape and properties of this distribution, see the Binomial Distribution Calculator. For a single trial's odds, the Probability Calculator covers basic probability rules.