Formula and units
X ∼ Binomial(n, p); P(X = k) = binom(n, k) pᵏ(1 − p)ⁿ⁻ᵏ
binom(n, k) = n! / [k!(n − k)!]
E[X] = np people; E[min(X, d)] = Σₖ₌₀ⁿ min(k, d) P(X = k) desks
q = P(X > d) = Σₖ₌d₊₁ⁿ P(X = k); expected short days = 20q
people per desk = n / d; n* = max {n ∈ integers ≥ 0 : P(Binomial(n, p) > d) ≤ α}
n is a whole number of employees or members, d a positive whole number of shared desks, and k the number attending. p is each person’s daily attendance percentage divided by 100; α is the target percentage of short days divided by 100. q is the probability of any shortage on a day. Probabilities have no units; n/d is people per desk. An empty sum when d ≥ n is zero.
Worked example
With n = 10 people, d = 5 desks and p = 0.5, the ratio is 2 people per desk and mean attendance is 5 people. P(X > 5) = 386/1024 = 37.70%, giving 20 × 386/1024 = 7.54 expected short days per month. Mean desks used is 4490/1024 = 4.38, capped by availability. For a target α = 0.05 with those same 5 desks and attendance probability, at most 6 people can share: 6 people give 1/64 = 1.5625% short days, while 7 give 8/128 = 6.25%.
Assumptions and limits
- Everyone has the same fixed attendance chance, makes the decision independently, and keeps one desk all day. Team days break independence and peak weekdays can raise the risk. Use actual attendance patterns rather than assuming the weekly average applies every day.
- Part-day visits let one desk serve successive people. The articles simulate arrival and departure times; their fraction of visits without a desk differs from the fraction of days with any shortage here. Dedicated desks, guests, preferences and reservations are not extra capacity in this model.
- The calculator sums binomial probabilities with a recurrence scaled at the mode, without a normal approximation. Its integer search ends at 1,000,000 people and reports when the maximum is unresolved. With positive attendance a 0% shortage target allows at most d people; 0% attendance or a 100% target has no finite maximum. Twenty working days is an assumption, and an expected count is not a monthly guarantee.