When to use this
Use this probability model when the setup matches
Coin flips, pass-fail trials, repeated yes-no outcomes, and any fixed number of independent trials with a constant probability p.
What this result means
Interpret the probability in three formats
The result gives the chance of the requested success count range. It works when each trial behaves the same way and one trial does not change the probability of the next one.
Formula: P(X = k) = C(n, k)p^k(1-p)^(n-k)
Inputs
Describe the trial count, target successes, and success probability
This calculator uses the binomial model, so each trial must be independent and use the same probability p.
Number of trials
How many repeated trials or flips you perform.
What to enter: A whole number 0 or larger.
How to use it: Use the full count of repeated identical trials.
Example: Five coin flips means 5 trials.
Target successes
The number of successes you are asking about.
What to enter: A whole number between 0 and the number of trials.
How to use it: Pair this with exactly, at least, or at most.
Example: Exactly 3 heads means 3 target successes.
Success probability
The probability of success on one trial.
What to enter: Use a decimal like 0.5, a fraction like 1/2, or a percent like 50%.
How to use it: Keep the same value for every trial.
Example: A fair coin has p = 0.5 for heads.
Event type
Choose exactly, at least, or at most.
What to enter: Pick the wording that matches the question.
How to use it: At least and at most add several exact-success terms together.
Example: At least 1 head in 4 flips sums the probabilities for 1, 2, 3, and 4 heads.
Worked examples
Quick checks with common probability questions
Exactly 3 heads in 5 flips
A fair coin flip model with 5 independent trials and p = 0.5.
The probability of exactly 3 successes is 31.25%.
Load this example into the calculatorAt least 1 head in 4 flips
This sums the probabilities for 1, 2, 3, and 4 heads.
The probability of at least 1 success is 93.75%.
Load this example into the calculator