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Discrete Probability Calculator

1. Define Dimensions

2. Input Probabilities

Enter values for P(X,Y).

Total Probability: 0.00

3. Calculate

Σ How I Analyze Joint Probability

In my probability course I encountered a difficult problem that required looking at two variables at the same time. I was analyzing a software program that had two different modules. I needed to track the number of errors in Module 1 (variable X) versus the number of errors in Module 2 (variable Y). That assignment taught me that looking at one variable in isolation often hides the real story. I built this calculator to handle those complex "X and Y" scenarios where you need to check if two events are truly independent or if they influence each other. For example I had to calculate if a bug in the first module made it more likely to find a bug in the second module.

Step-by-Step Instructions:

  1. Set your grid size - Enter the number of rows (X values) and columns (Y values), then click "Generate"
  2. Define your variables - Fill in the X and Y value headers in the grid
  3. Enter probabilities - Fill in each cell with P(X,Y). Use decimals (0.25) or fractions (1/4)
  4. Verify your sum - Make sure all probabilities add up to exactly 1.0
  5. Calculate - Click "CALCULATE RESULTS" to see marginal distributions, covariance, and correlation

Poisson distribution guide

Counting independent events in a fixed interval

The Poisson distribution models how many times an event occurs in a specified period, area, distance, or volume. Its single parameter, λ (lambda), is the expected number of events in that exact interval.

Assumptions to check

  • Events occur independently.
  • The average rate stays constant throughout the chosen interval.
  • Two events do not occur at precisely the same infinitesimal instant.
  • The interval used for λ matches the interval in the question.

If calls arrive at four per hour, λ is 4 for one hour and 2 for thirty minutes. Mixing rates and interval lengths is one of the most common Poisson-model errors.

Formula and key properties

For exactly k events, P(X = k) = e-λλk/k!, where k is a nonnegative integer. Both the mean and variance equal λ, and the standard deviation is √λ.

That mean-equals-variance property is a useful diagnostic. Real count data with much greater variance may contain clustering or changing rates; data with much smaller variance may reflect spacing or capacity constraints.

Worked arrival example

A help desk receives an average of four independent requests per hour. For exactly six requests in the next hour, set λ = 4 and k = 6.

P(X = 6) = e-446/6! ≈ 0.1042, or 10.42%. The probability of zero requests is e-4 ≈ 0.0183. Although four is the expected count, values above and below four remain possible and are visible in the calculator's probability curve.

Poisson versus binomial

A binomial experiment begins with a fixed number of opportunities and a success probability. A Poisson experiment begins with an exposure interval and an average event rate, with no fixed maximum event count.

Poisson can approximate a binomial model when n is large and p is small, using λ = np, but the exact binomial calculation is preferable when n and p are known and computationally manageable.

Limitations and reference

Do not use a single Poisson rate when events influence one another, demand changes sharply during the interval, or the process has a hard capacity. For quality-control examples and the relationship to rare binomial events, see the NIST/SEMATECH count-model reference. For a fixed trial count, use the Binomial calculator.