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Poisson Distribution Calculator
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Enter the average rate (λ) and number of events (k) to quickly calculate the exact Poisson probability P(X=k) and cumulative probabilities.
λ is the average number of events in a given time/space interval, can be a decimal, and must be ≥ 0. k is the actual number of occurrences you want to calculate, and must be a non-negative integer.
Enter the average rate λ and number of occurrences k to view probabilities.
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If a call center receives an average of 5 complaint calls per hour, what is the probability of receiving exactly 0 calls between 10 AM and 11 AM next Monday? Or if a hospital emergency room averages 1.2 patients every 10 minutes, what is the probability that exactly 3 patients will arrive in a specific 10-minute window? These scenarios are perfect for estimation using the Poisson distribution. The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, introduced by French mathematician Siméon Denis Poisson in 1837. Its key assumptions are: events occur independently of each other, and the average rate (λ) is constant.
In our calculator, you only need to enter two numbers: λ (the average rate of occurrence) and k (the number of events you want to check). The tool will calculate the exact probability of exactly k events occurring, as well as the cumulative probabilities (≤k and ≥k).
The Poisson probability formula is simple: P(X=k) = (λk × e-λ) / k!
Meaning of each variable:
Why does the formula look like this? Simply put: the Poisson distribution is the limit of the binomial distribution when the number of trials is very large and the probability of success is very small. λ represents the expected value, e-λ ensures that the sum of all probabilities equals 1, and λk/k! increases then decreases as k grows, forming a unimodal shape overall.
The calculator also provides cumulative probabilities: P(X ≤ k) = the sum of all probabilities up to k, and P(X ≥ k) = 1 - P(X ≤ k-1). You can see these three values on the results card on the right.
Open our Poisson Distribution Calculator, and you will see two input fields:
Note: Both λ and k must be numbers ≥ 0, and k must be an integer. If you enter a decimal for k, the calculator will automatically round it to the nearest integer.
Let's demonstrate with a real-world scenario: A customer service center receives an average of 4 customer calls per hour (λ=4). What is the probability of receiving exactly 2 calls in a given hour?
Step 1: Enter 4 in the λ input field and 2 in the k input field.
Step 2: Click calculate. The calculator will plug the values into the formula: P(X=2) = 42 × e-4 / 2! = 16 × 0.018316 / 2 = 0.1465. The result shows: P(X=2) ≈ 0.1465 (about 14.65%).
Step 3: You will also see the cumulative probabilities: P(X≤2) ≈ 0.2381 (23.81%), and P(X≥2) ≈ 0.9084 (90.84%).
Interpretation: Even though the average is 4 calls per hour, the probability of getting exactly 2 calls is not very high (14.65%) due to variance. The highest probabilities occur when k is 3 or 4 (around λ).
Comparison Example (Extreme Values): What if λ=0.2 and k=0 (a rare event, like occurring once every 5 hours)? Enter 0.2 and 0, and you get P(X=0) = e-0.2 ≈ 0.8187 (81.87%). This shows that at very low occurrence rates, the event will not happen at all most of the time. Conversely, if λ=10 and k=15, P ≈ 0.0347 (3.47%), which is much smaller than the probabilities near λ.
| Probability Range | Meaning |
|---|---|
| P(X=k) > 0.1 | This number of occurrences is very common and is close to the average rate. |
| 0.01 < P(X=k) ≤ 0.1 | Not rare, but still possible. |
| P(X=k) ≤ 0.01 | A low-probability event, usually considered worthy of attention (e.g., an anomaly in quality control). |
| P(X ≤ k) is very small (e.g., <0.05) | The current k is far below average, indicating an unusually "cold" scenario. |
| P(X ≥ k) is very small (e.g., <0.05) | The current k is far above average, indicating a "burst" or spike scenario. |
In everyday use, a threshold of 0.05 is often used to determine if something is anomalous (similar to a p-value in statistical testing).
1. Call Center Scheduling: A customer service manager uses λ = average hourly calls to calculate P(X≥10) to assess if staffing is sufficient and avoid call overflow.
2. Factory Yield Monitoring: A quality inspector knows the product defect rate is λ = 2 per 1,000 units. When checking a batch of 1,000 units, if k = 6 defects are found, they calculate P(X≥6) to determine if this is an abnormal fluctuation and decide whether to halt production for troubleshooting.
3. Traffic Accident Analysis: A specific road segment averages 3 accidents per month. Traffic police calculate P(X=0) to predict the likelihood of a "zero-accident month" to allocate patrol resources accordingly.
4. Library Checkout Prediction: A library checks out an average of 15 books every half hour. They want to know the probability of checking out >20 books in a given half hour to decide whether to open more service desks.
Yes. λ can be any non-negative real number, such as 0.5 or 2.7. The calculator uses floating-point arithmetic internally, and results are accurate to 4 decimal places.
In our calculator, k is supported up to 999. Because factorial calculations yield extremely large numbers, when λ is too large (e.g., λ > 200), the exact probability might be so small that it displays as 0.0000, but the cumulative probabilities remain valid.
When λ is large (generally λ ≥ 20), the Poisson distribution approximates a normal distribution, with both mean and variance ≈ λ. In this case, you can use a normal approximation for quick estimation, but the Poisson distribution is more precise.
Plugging k=0 into the formula: P(X=0) = λ0 × e-λ / 0! = 1 × e-λ / 1 = e-λ. So, as long as you know λ, you can instantly calculate the probability of the event not happening at all.
The Poisson distribution is typically used for "rare events". If the number of trials n is large and the probability p is small (usually np ≤ 10), you can approximate it using λ = np. Simply enter this λ into the calculator. For an exact binomial distribution, please switch to our Binomial Distribution Calculator.
It shouldn't be. A cumulative probability is the sum of mutually exclusive probabilities, so the total must be ≤ 1. If you see a value greater than 1, you may have entered a negative or invalid number. Please check your λ and k inputs.
Now you can try your own numbers in the calculator above—for example, if an average of 30 customers walk past your store every day, and you want to know the probability of exactly 25 customers passing by, enter λ=30 and k=25 to see the result!
