AP Stats Poisson Distribution Calculator
The Poisson distribution models how many times an event happens in a fixed interval, when events happen independently at a known average rate. If a call centre gets an average of 3 calls per hour, the Poisson distribution gives the probability of exactly 2 calls, or at most 2 calls, in an hour.
Poisson Distribution Calculator
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What Is the Poisson Distribution?
The Poisson distribution gives the probability of a given number of events happening in a fixed interval of time or space, when events happen independently at a known average rate. For example, if a call center gets an average of 4 calls per hour, the Poisson distribution gives the chance of exactly 6 calls in the next hour. Its mean and its variance are both equal to the average rate.
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Example: lambda = 3, x = 2 — P(X = 2) = 0.2240.
How to Use This Calculator
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Step 1: Open the AP Stats Poisson Distribution Calculator calculator at the top and enter the values it asks for.
1. Enter lambda, the average rate — for example, 3 calls per hour, or 2.5 accidents per week. 2. Enter x, the count you want the probability for. 3. Press Calculate. 4. Read P(X = x) for exactly that count, and P(X ≤ x) for that count or fewer. The bar chart shows the whole distribution, with your x highlighted.
How to Calculate a Poisson Probability
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The formula is P(X = x) = e^(−λ) × λ^x / x!, where e ≈ 2.71828 and x! is the factorial of x. For a cumulative probability, add the probabilities for every count from 0 up to x: P(X ≤ x) = P(X = 0) + P(X = 1) + … + P(X = x).
Worked example: Lambda = 3, x = 2. Then P(X = 2) = e^(−3) × 3² / 2! = 0.0498 × 9 / 2 = 0.2240. The cumulative probability is P(X ≤ 2) = P(0) + P(1) + P(2) = 0.0498 + 0.1494 + 0.2240 = 0.4232. So there is about a 22.4% chance of exactly 2 events, and about a 42.3% chance of 2 or fewer.
Tips and Common Mistakes
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Mistake: Using a binomial with a fixed n for rare events over time.
- Match lambda and x to the same interval. If lambda is per hour, x must also be a count per hour.
- Do not use Poisson for a fixed number of trials with a success probability — that is binomial. Poisson is for counts in time or space with no fixed trial count.
- For “at least x,” use 1 − P(X ≤ x − 1), not 1 − P(X = x).
- Remember that for Poisson, the mean and the variance are both lambda.
- Check the conditions: events occur independently, at a constant average rate, and two events do not happen at exactly the same instant.
Frequently Asked Questions
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When should I use Poisson instead of binomial?
Use Poisson for counts over time or space with no fixed number of trials, such as calls per hour or typos per page. Use binomial when there are a fixed number of trials, each with the same success probability.
What is lambda in a Poisson problem?
Lambda is the mean number of events in the interval given in the question. It is also the variance of the distribution, and the standard deviation is √λ.
How do I find P(X ≥ x) with this calculator?
Find P(X ≤ x − 1) here, then subtract from 1. For example, P(X ≥ 3) = 1 − P(X ≤ 2).
Can lambda be a decimal?
Yes. Lambda can be any positive number, such as 2.5 events per week. Only x must be a whole number, because it is a count.
What does the shape of a Poisson distribution look like?
For small lambda it is right-skewed, with a peak near lambda. As lambda grows, it becomes more symmetric and looks more like a normal distribution.
Is Poisson on the AP Statistics formula sheet?
Poisson is less central than normal and binomial in AP Statistics, but the same counting and probability ideas apply, and the formula here is the standard one used in statistics courses.
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Poisson Distribution Calculator → Binomial Distribution Calculator → Geometric Distribution Calculator