AI Grounds

Open AI Grounds on a desktop

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Guided discovery

Waiting & Arrival Distributions

Change the event chance. Separate waiting time from arrival count.

One-second model: independent ticks, the same event chance p each second, at most one event per tick. Scenarios are illustrative parameter choices.

Chance on each independent tick, not events per second. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.

How long to count events. The waiting law keeps the same p. Use arrow keys on the slider. Press Enter or leave the number field to apply an exact edit.

Waiting for first event

Geometric: exact tick probabilities. Wait counts the successful second; it can be longer than this window.

Fixed scale: 0–100% probability

  • 1–10 s≈ 18.29%
  • 11–20 s≈ 14.95%
  • 21–30 s≈ 12.21%
  • 31–40 s≈ 9.98%
  • 41–50 s≈ 8.15%
  • 51–60 s≈ 6.66%
  • > 60 s≈ 29.76%

Counts in 5 minutes

Poisson count approximation: most useful for small per-second p. 9+ pools all counts nine and above, not one outcome.

Fixed scale: 0–100% probability

  • 0≈ 0.25%
  • 1≈ 1.49%
  • 2≈ 4.46%
  • 3≈ 8.92%
  • 4≈ 13.39%
  • 5≈ 16.06%
  • 6≈ 16.06%
  • 7≈ 13.77%
  • 8≈ 10.33%
  • 9+≈ 15.28%

P(wait ≤ 20 s): ≈ 33.24% · P(wait > 60 s): ≈ 29.76% · Median wait: 35 s. Median is the earliest tick with at least 50% chance of an event by then.

Mean wait

Average first-event wait over many independent runs.

50 s

1 / p = 1 / 0.02

An average, not a scheduled arrival.

Expected count

Average events in 5 minutes; both models share this mean.

6

μ = 60 × 5 × 0.02

A count can exceed one or be fractional.

Rate

Expected events per minute at the current chance.

1.2/min

λ = 60p events/minute

Changing Window keeps this rate fixed.

Chance of at least one event in this window

Two different approximations: small p supports Poisson counts; small μ (rate × time) supports the linear probability shortcut.

Exact tick model
≈ 99.77%

1 − (1 − p)300

Poisson approximation
≈ 99.75%

1 − e−μ · exact for Poisson, approximate for ticks

Linear shortcut: μ
600%

Outside the probability range. An expected count is not a probability; this value is not clipped.

Sample tick timeline

One illustrative pseudorandom run: one draw per second with fixed seed 3209. Actual count: 5; expected count: 6. Changing p reuses draws; extending Window preserves earlier events. A sample is not a probability distribution.

0 s300 s

Event seconds: 169, 171, 205, 212, 221. The window endpoint can cut off a wait; it is not treated as a completed inter-event gap.

Formula details

W is the integer tick of the first event; K is the number of events in the window. N = 60 × Window seconds; μ = Np. All numbers shown above are rounded for display.

  • Geometric: P(W = k) = (1 − p)k−1 p, k = 1, 2, …; P(W > k) = (1 − p)k.
  • Exact tick count is binomial: N independent trials with the same p. P(K = 0) = (1 − p)N; P(K ≥ 1) = 1 − P(K = 0).
  • Poisson approximation: P(K = k) ≈ e−μ μk/k!, k = 0, 1, … . k! multiplies integers 1 through k; 0! = 1. The 9+ bar is one minus the sum of counts 0 through 8.
  • The linear shortcut 1 − e−μ ≈ μ needs μ much smaller than one. It can fail even with rare per-second events if the window is long.

Chance 2% per second; window 5 minutes; mean wait 50 seconds; expected count 6. At least one event: exact tick ≈ 99.77%, Poisson approximation ≈ 99.75%, linear shortcut 600%, outside probability range.