Binomial Percent Point

Find the smallest integer k whose cumulative probability reaches q, from probability q, trials n, and success probability p.

Input

Enter probability q, trials n, and success probability p to find the smallest integer k where the cumulative probability P(X is at most k) reaches q.

Probability q

Enter a value greater than 0 and at most 1 (e.g. 0.95).

Number of trials n

Enter an integer of 1 or more (e.g. 20).

Success probability p

Enter a value from 0 to 1 (e.g. 0.5).

Percent point k for q=0.95, n=20, p=0.5

Cumulative P(X is at most k)

0.97930527

Mass P(X equals k)

0.03696442

Upper P(X is at least k)

0.05765915

Mean

Variance

Probability mass P(X equals k)

Cumulative probability P(X is at most k)

The binomial distribution models the number of successes X in n independent trials, each with success probability p. The probability mass is P(X = k) = C(n, k) times p^k times (1 minus p)^(n minus k), for integer k from 0 to n.
The percent point is the smallest integer k for which the cumulative probability P(X is at most k) is greater than or equal to the input probability q. Because the distribution is discrete, k is integer valued and may not match q exactly.
The cumulative probability is the sum of P(X = j) for j from 0 to k. This tool accumulates the mass from k equals 0 upward and returns the first k whose running total reaches q.
The mean equals n times p and the variance equals n times p times (1 minus p). The charts show the probability mass and cumulative probability for each k, highlighting the resulting percent point.
The binomial coefficient C(n, k) is evaluated in the logarithmic domain using the log gamma function ln of Gamma, rather than direct factorials, so it stays stable and avoids overflow even for large n.

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