Why Do Three Buses Arrive at Once?
By Numeric Forest Team | Published on 08 May 2026
It is common to observe long gaps between events, followed by several events occurring close together. A typical question is why several buses, deliveries, or messages sometimes appear at almost the same time. One way to explore this behaviour is to model the number of events using the Poisson Distribution.
This article provides an instructional example showing how the Poisson Distribution Calculator can be used to examine the probability of observing exactly three events, and how likely it is to see different event counts in a given setting.
Modelling random event counts
When events occur independently and at a roughly steady average rate, the Poisson distribution is often used to describe how many events may occur in a fixed interval. The model does not describe any specific transport system or service; it is a general statistical framework for counting events.
In this context, an "event" could represent a bus arrival, a delivery, a message, or any other occurrence that can be counted. The Poisson distribution focuses on the number of events, given an assumed average rate.
Inputs used in the Poisson Distribution Calculator
The Poisson Distribution Calculator uses several inputs to define the model and the required probability. The main inputs are:
- Average rate (λ): the expected number of events in the interval being considered. In simple terms, this is how many events tend to occur on average.
- Probability type: specifies whether the probability is for exactly a given event count, less than, greater than, or between two counts.
- Number of events (x): the event count of interest, such as exactly three events.
- Upper bound (x₂): an additional event count used when calculating the probability between two values.
- Output type: selects whether the results are shown as a chart or as a table.
- Decimal places: controls how many decimal places are used when displaying numerical results.
These inputs allow the calculator to present probabilities for different event counts under the assumptions of the Poisson model.
Example: Probability of exactly three events
To provide a concrete example, consider a situation where the average rate of events is set to 5. The calculator can then be used to determine the probability of observing exactly 3 events.
Average rate (λ) = 5
Probability type = Equal P(X = x)
Number of events (x) = 3
Upper bound (x₂) = 5 (used only for "between" probabilities)
Output type = Chart (table can also be selected)
Decimal places = 2
With these values, the calculator evaluates the probability of exactly three events occurring under the Poisson model with an average rate of 5. The resulting probability is approximately 14.04% when rounded to two decimal places.
Distribution of event counts from 0 to 5
The Poisson Distribution Calculator can also display how likely it is to observe different event counts using the same average rate. The following table shows the probability of observing between 0 and 5 events when the average rate is 5, rounded to two decimal places.
| Number of events (x) | Probability |
|---|---|
| 0 | 0.67% |
| 1 | 3.37% |
| 2 | 8.42% |
| 3 | 14.04% |
| 4 | 17.55% |
| 5 | 17.55% |
These values illustrate that, for this particular average rate, several different event counts have non-negligible probabilities. The calculator can present these results either as a chart or as a table, depending on the selected output type.
Using the calculator for further exploration
The same approach can be applied to many everyday counting problems, such as the number of arrivals, messages, or other events in a given period. By adjusting the average rate and the event count in the Poisson Distribution Calculator, it is possible to examine how the probability changes under different assumptions.
This example is intended to illustrate the use of the Poisson model in a simplified setting. Real-world systems may involve additional factors and complexities that are not captured by this basic framework.
Interactive analysis
To explore other scenarios, the Poisson Distribution Calculator may be used to compute probabilities for different average rates, event counts, and probability types. The chart and table options provide alternative ways to view the distribution of event counts.
Disclaimer: This article is for informational and educational purposes only. It does not describe any specific transport system or operational service and should not be used for planning, prediction, or decision-making. Real-world systems may differ significantly from the simplified models discussed here.