Calculator Input
Enter comma-separated values for counts, probabilities, and optional labels.
Example Data Table
This sample shows a valid multinomial setup with three categories.
| Category | Observed Count | Probability | Expected Count |
|---|---|---|---|
| Red | 4 | 0.40 | 4.00 |
| Blue | 3 | 0.35 | 3.50 |
| Green | 3 | 0.25 | 2.50 |
Formula Used
The multinomial probability mass function gives the probability of a specific count pattern across several categories.
P(X = x) = n! / (x1! x2! ... xk!) × p1x1 p2x2 ... pkxk
Conditions:
x1 + x2 + ... + xk = n
p1 + p2 + ... + pk = 1
Expected count for category i: E[Xi] = n × pi
Variance for category i: Var(Xi) = n × pi × (1 − pi)
This calculator uses logarithms internally for numerical stability when probabilities become very small.
How to Use This Calculator
- Enter the total number of trials.
- Type category counts as comma-separated integers.
- Type category probabilities in the same order.
- Optionally add category labels.
- Choose decimal precision for the output.
- Click Calculate PMF.
- Review the PMF, coefficient, table, and graph.
- Use CSV or PDF export for reporting.
Frequently Asked Questions
1. What does this calculator measure?
It computes the probability of one exact count pattern under a multinomial model. It also shows expected counts, variance, standard deviation, and a chart for quick interpretation.
2. When should I use a multinomial distribution?
Use it when each trial has one outcome from several categories, the category probabilities stay constant, and trials are independent. Survey responses, product choices, and defect types are common examples.
3. Why must counts sum to total trials?
A multinomial outcome splits all trials across categories. Because every trial belongs to exactly one category, the total of category counts must match the full number of trials.
4. Why must probabilities sum to one?
The probabilities represent all possible category outcomes for a single trial. Since one of those outcomes must occur, their combined probability must be exactly one.
5. Can the calculator handle many categories?
Yes. Add matching comma-separated counts, probabilities, and labels. The only requirement is consistent list length, valid probabilities, and a count total equal to the trial count.
6. Why is logarithmic calculation used internally?
Exact multinomial probabilities can become extremely small. Logarithms reduce underflow and improve numerical stability, especially when the trial count is large or many categories are included.
7. What do expected counts tell me?
Expected counts show the average category totals predicted by the chosen probabilities over many repeated experiments. They help compare your observed counts with the model's central tendency.
8. What is the difference between binomial and multinomial?
A binomial model has two outcomes per trial. A multinomial model extends that idea to three or more categories while keeping one outcome per trial.