How this calculation works
The exponential distribution models a nonnegative continuous waiting time with a constant hazard. Multiply the rate by the waiting time using matching units, then calculate the cumulative and remaining tail probabilities.
P(T ≤ t) = 1−e⁻λt; expected wait = 1/λA worked example
At 0.5 events per minute, the chance of waiting at most 2 minutes is 1−e⁻¹, approximately 63.212056%.
Good to know
A mathematical probability model, not a fit test or an empirical guarantee. Inputs must match the model assumptions. Floating-point arithmetic is used; extremely small probabilities may underflow or round to zero.
Reference: NIST: exponential distribution (opens in a new tab) ↗
Use one time unit consistently
A rate of 0.5 events per minute and a wait of two minutes gives λt = 1. Converting both to hours gives 30 events per hour and 1/30 hour, with the same product and probability. Converting only one input changes the question.
The mean waiting time is the reciprocal rate. At 0.5 per minute it is two minutes. The chance that the next event arrives by that mean wait is about 63.21%, so the mean is not the same as the 50th percentile.
The elapsed-time probability and the remaining tail are complements. At t = 0, the cumulative probability is zero; for longer nonnegative times, it increases toward one.
Memorylessness is an assumption with consequences
In an exponential model, waiting longer already does not alter the distribution of the remaining wait. After five event-free minutes, the next two minutes have the same event probability as any fresh two-minute interval under the same constant rate.
This can be reasonable for an ideal constant-rate process. It is often a poor model when aging, wear, a schedule or changing demand affects the event rate. A bus scheduled to arrive at a particular time is not made memoryless by entering an average interval.
The rate must be positive and the wait nonnegative. If no events can ever occur, an ordinary exponential distribution with finite mean is not the appropriate model. A tiny positive rate is different from a guaranteed event-free process.
A few questions, answered
Is this a Poisson event-count calculation?
It is the associated waiting-time model for an ideal homogeneous Poisson process. The Poisson calculator counts events in a fixed interval instead of timing the next event.
What is the probability of one exact wait?
In a continuous model, an exact point has probability zero. This tool evaluates cumulative waits and their complements.
Can I use minutes, hours or days?
Yes. The units are your choice, but the event rate and waiting time must use the same time unit.