Exponential Distribution Calculator (Waiting Times)
Chance that the next event — a customer, a failure, a call — comes within a given time when events arrive at random at a known average rate.
Results
Chance it happens within that time
39.347
%
Chance of waiting longer than that
60.653
%
Rate (events per unit of time)
0.10000
Median wait
6.9315
Wait that covers 90 % of cases
23.0259
The dashed line marks the value you entered.
What this tool does
When events happen independently at a steady average rate, the wait until the next one follows the exponential distribution. Its odd property is that it has no memory: having already waited ten minutes does not make the next minute any more likely to bring the bus. Half of all waits are shorter than 69 % of the average, but one in ten is longer than 2.3 times the average — which is why random arrivals feel so uneven. Use any unit of time, as long as both fields use the same one.
Formula
P(T ≤ t) = 1 − e^(−t ÷ μ) · λ = 1 ÷ μ · median = μ ln 2
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mt | Average time between events | — |
x | Waiting time | — |
PL | Chance it happens within that time | % |
PG | Chance of waiting longer than that | % |
RT | Rate (events per unit of time) | — |
MD | Median wait | — |
P9 | Wait that covers 90 % of cases | — |
Worked example
- Average time between events10
- Waiting time5
- Chance it happens within that time39.347 %
- Chance of waiting longer than that60.653 %
- Rate (events per unit of time)0.10000
- Median wait6.9315
- Wait that covers 90 % of cases23.0259
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.