Poisson probability formula: P(X = k) arrivals with EXP and FACT

Model independent arrival counts per hour, day, or square meter

While Binomial models k successes out of a fixed n trials, the Poisson distribution models k discrete events occurring in a fixed window of time or space when the average rate is lambda (e.g., lambda = 2.0 emergency arrivals/hour or 0.5 server failures/month).

Using universal functions EXP and FACT, exact Poisson PMF is P(X = k) = (lambda^k * EXP(-lambda)) / FACT(k)! Even better: probability of ZERO events is simply EXP(-lambda), so probability of AT LEAST ONE event is 1 - EXP(-lambda)!

Exact Poisson PMF P(X = k) and P(X >= 1) formulas

This bounded factorial demonstration validates the count and handles lambda=0 explicitly. Larger parameters need a numerically stable distribution implementation.

=IF(OR(A2<0,A2>100,B2<0,B2>100,B2<>INT(B2)),"Outside example domain",IF(A2=0,IF(B2=0,1,0),ROUND(((A2 ^ B2) * EXP(-A2)) / FACT(B2), 4)))

Probability of at least 1 event occurring in the window

Complementary probability 1 - P(X = 0) = 1 - EXP(-lambda).

=ROUND(1 - EXP(-A2), 4)

Inputs and interpretation

The Poisson model assumes independent counts in disjoint intervals and an appropriate constant event rate for the window. Lambda is the expected count in that window, not a rate with unmatched time units. This elementary factorial demonstration is bounded to lambda and integer k from 0 through 100 to avoid overflow; rounded results may show small nonzero probabilities as zero.

Further reading

• Microsoft Support — EXP function
• Microsoft Support — FACT function
• NIST — Poisson distribution