Poisson Distribution Calculator
The Poisson Distribution Calculator allows you to compute both individual and cumulative Poisson probabilities with ease. Use it to solve real-world problems involving rare events, error rates, arrivals, or occurrences over a fixed interval.
■ Select a probability from the dropdown box.
■ Enter values for μ and x (in the first two textboxes).
■ Click Calculate to compute probabilities.
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Probability Distribution
| Number of successes, x | P(X = x) | P(X ≤ x) |
|---|---|---|
| – | – | – |
Probability Distribution Graph
Distribution Details
MEAN (μ)
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STD DEV (σ)
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INSIGHT
Select parameters and click Calculate to see probability insights.
What is Poisson Distribution?
- Poisson distribution measures the probability of events occurring in a fixed interval.
- Used for rare events like phone calls, customer arrivals, or defects.
- Formula: P(X = x) = (e^(-μ) × μ^x) / x!
Real-World Examples
- Receptionist: Average 2 calls/hour. P(exactly 5 calls)?
- Quality Control: Average 1.5 defects per 1000 items. P(3 defects)?
- Customer Service: Average 10 customers/hour. P(at least 12)?
Understanding Results
- P(X = x): Exact probability of exactly x events.
- P(X ≤ x): Probability of x or fewer events (cumulative).
- P(X ≥ x): Probability of x or more events.
- Standard Deviation: σ = √μ (spread of results)
Typing Errors Scenario
An expert typist makes an average of 2 typing errors every 5 pages. What is the probability that the typist will make at most 5 errors on the next 15 pages?
- Identify known values:
| Parameter | Value | Explanation |
|---|---|---|
| Poisson random variable (x) | 5 | We want the probability of at most 5 errors |
| Average rate (λ) | 6 | 15 pages is 3× the original 5 pages, so 3 × 2 errors = 6 |
- Select
P(X ≤ x)from the calculator. - Enter λ = 6 and x = 5 into the Poisson Calculator.
- Click Calculate.
Calculator Output
| Probability Type | Result |
|---|---|
| P(X ≤ 5) (At most 5 errors) | 0.44568 |
| P(X = 5) (Exactly 5 errors) | 0.16062 |
Interpretation:
There is a 44.568% chance that the typist will make five or fewer errors on 15 pages.
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