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Event{{ eventShortLabel }} Expected{{ resultsReady ? formattedMean : '—' }} Model fitUser confirmed
Choose the event wording before entering its count bound.
A whole number from 0 to 10,000, fixed before the experiment.
Enter a decimal from 0 to 1, such as 0.25 for 25%.
{{ event_mode === 'between' ? 'The inclusive lower bound of the interval.' : 'The inclusive threshold for the selected event.' }}
A whole number from the lower bound through n.
The neutral default is 8 significant digits.
Event analysis
{{ formattedProbability }}
{{ formattedPercent }} of total probability mass
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Calculation method
Inclusive events sum the relevant probability mass function terms.
P(X=k)=(nk)pk(1−p)n−k
  1. {{ eventLongLabel }}
  2. Selected event mass: {{ formattedProbability }}
  3. Complementary mass: {{ formattedComplement }}
Confirm the model before using the result
  • The trial count is fixed in advance.
  • Each trial has two outcomes under one success definition.
  • Trials are independent and use the same success probability.
This tool evaluates the declared model. It does not verify these assumptions or establish a statistical conclusion.

The chart renderer is unavailable. The same values remain available in Event analysis and the calculation ledger.

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Repeated trials often lead to a count rather than a single yes-or-no result. A batch may contain seven defective parts, a survey may receive twelve favorable answers, or ten independent attempts may produce exactly four successes. The binomial distribution assigns probabilities to those success counts when the experiment follows one stable model.

Four conditions define that model:

  • The number of trials is fixed before the experiment.
  • Each trial has two mutually exclusive outcomes, labeled success and failure.
  • Trials are independent, so one outcome does not change another trial's chance.
  • The success probability stays constant across every trial.

“Success” is only the event being counted. It can mean a sale, a failed component, a positive test, or any other chosen outcome. Its probability must be entered as a decimal from 0 to 1, so 25% becomes 0.25.

Event wording Included success counts
Exactly k Only k
At most k 0 through k, including both ends
At least k k through n, including both ends
Between a and b a through b, including both ends

A precise probability does not repair a poor model. Sampling without replacement from a small population, changing success rates, clustered outcomes, or feedback between trials can violate the binomial assumptions. In those settings, a hypergeometric, Poisson-binomial, beta-binomial, or simulation approach may fit better.

How to Use This Tool:

Define the event first so the entered count has the intended inclusive boundary.

  1. Choose Probability event: exactly, at most, at least, or between.
  2. Enter Number of trials (n) as a whole number from 0 through 10,000 and Success probability per trial (p) as a decimal from 0 through 1.
  3. Enter the requested success count. For a between event, also enter the inclusive upper count, which must be at least the lower count and no greater than n.
  4. Use Displayed significant digits to change presentation precision only. The underlying probability is not recalculated from the rounded display.
  5. Read the requested probability beside its complement, then use the mean and standard deviation to understand where the count sits within the model.

Interpreting Results:

The requested event probability and complement should sum to 1 apart from display rounding. The complement covers every valid success count outside the selected event. For an inclusive interval, that means counts below a or above b.

Expected value is the long-run average success count across many repetitions of the full experiment, not a promise for the next run. Standard deviation describes typical spread in count units. Variance is the squared spread and therefore uses count².

Small probabilities can still occur, and large probabilities can still fail to occur on one run. Check the assumptions and the event wording before treating the percentage as evidence that an observed result is surprising.

Technical Details:

A binomial random variable X counts successes across n independent Bernoulli trials with constant success probability p. Its support contains only the integers from 0 through n.

Formula Core:

The probability mass function gives one exact count. Cumulative and interval events add the relevant exact-count terms.

P(X=k) = (n!k!(n-k)!) pk (1-p)n-k P(a≤X≤b) = ∑ k=a b P(X=k) E[X] = np Var(X) = np(1-p) σ = np(1-p)

For 10 trials with p = 0.5, exactly 7 successes has probability 0.1171875, or 11.71875%. The mean is 5, variance 2.5 count², and standard deviation about 1.58114 successes.

Rule Core:

Mode Selected sum Complement
Exactly k k only 0 to k - 1 and k + 1 to n
At most k 0 to k k + 1 to n
At least k k to n 0 to k - 1
Between a and b a to b 0 to a - 1 and b + 1 to n

For numerical stability up to 10,000 trials, factorial terms are handled in logarithmic form, range sums use scaled compensated addition, and the selected mass is normalized against selected plus complementary mass. Model outputs retain about 15 significant digits before the chosen display formatting.

Limitations:

The binomial distribution is inappropriate when trials influence one another, success probability changes, more than two outcomes must be modeled directly, or sampling without replacement materially changes the chance on later draws.

  • Counts and bounds must be whole numbers from 0 through n.
  • A between event includes both lower and upper bounds.
  • Changing significant digits changes only the displayed rounding, not the event probability.

References: