Statistical
Binomial probability formula: exact k successes in n independent trials
Calculate exact binomial probability P(X = k), expected successes n*p, and binomial standard deviation SQRT(n*p*(1-p)) using FACT.
Exact binomial probability from first principles
When each of n independent trials succeeds with probability p (and fails with 1 - p), the probability of observing exactly k successes is C(n, k) * p^k * (1 - p)^(n - k), where C(n, k) = n! / (k! * (n - k)!).
For n = 5 sales demos closing independently at p = 20% (0.20), the probability of closing exactly k = 2 deals is (120 / (2 * 6)) * 0.20^2 * 0.80^3 = 10 * 0.04 * 0.512 = 0.2048 (20.48%).
| Successes k (n=5, p=0.20) | Combinations C(5,k) | Exact P(X = k) |
|---|---|---|
| k = 0 | 1 | 0.3277 (32.77%) |
| k = 1 | 5 | 0.4096 (40.96%) |
| k = 2 | 10 | 0.2048 (20.48%) |
| k = 3 | 10 | 0.0512 (5.12%) |
Exact P(X = k), mean, and standard deviation formulas
The factorial demonstration is limited to integer trial counts from 0 through 100 to avoid large-factorial overflow. It handles probabilities 0 and 1 explicitly.
=IF(OR(A2<0,A2>100,A2<>INT(A2),B2<0,B2>A2,B2<>INT(B2),C2<0,C2>1), "Invalid inputs", IF(C2=0, IF(B2=0,1,0), IF(C2=1,IF(B2=A2,1,0),ROUND((FACT(A2) / (FACT(B2) * FACT(A2 - B2))) * (C2 ^ B2) * ((1 - C2) ^ (A2 - B2)), 4))))Binomial mean and standard deviation
Expected count equals n * p; standard deviation equals SQRT(n * p * (1 - p)).
=ROUND(SQRT(A2 * C2 * (1 - C2)), 4)Inputs and interpretation
Trials must be independent with the same fixed success probability. This is an exact-success count, not a cumulative probability. Rounded output may display very small nonzero probabilities as zero; large-n computations need a numerically stable distribution implementation.
Try the example
Use the original values or change them to test your understanding.