Calculator inputs
Enter the size of the sample space, the favorable outcomes, and optional repeated-trial settings. The grid adapts to three, two, or one column based on screen size.
Example data table
These sample scenarios show how theoretical probability changes with different sample spaces and repeated-trial assumptions.
| Scenario | Favorable Outcomes | Total Outcomes | Probability | Repeated Trials | At Least One Success |
|---|---|---|---|---|---|
| Even number on a die | 3 | 6 | 1/2 = 50% | 4 | 93.75% |
| Heart from a deck | 13 | 52 | 1/4 = 25% | 3 | 57.8125% |
| Prime number on a die | 3 | 6 | 1/2 = 50% | 2 | 75% |
| Red ball from 10 balls | 4 | 10 | 2/5 = 40% | 5 | 92.224% |
Formula used
1. Theoretical probability
P(E) = Favorable Outcomes / Total Possible Outcomes
2. Complement probability
P(E′) = 1 − P(E)
3. At least one success in n trials
P(at least one) = 1 − (1 − p)n
4. Exactly k successes in n trials
P(X = k) = C(n, k) × pk × (1 − p)n−k
5. Expected successes
Expected Value = n × p
The calculator starts with the classic theoretical probability ratio and then extends the analysis using binomial probability for repeated independent trials.
Odds in favor compare favorable outcomes to unfavorable outcomes, while odds against reverse that relationship. Fractions are simplified using the greatest common divisor.
How to use this calculator
- Type a short event name so the result is easier to read.
- Enter the number of favorable outcomes for the event.
- Enter the total number of possible outcomes in the sample space.
- Set the number of independent trials when the same probability repeats.
- Choose a target number of successes for repeated-trial analysis.
- Add the planned observations value to estimate expected successes.
- Press the calculate button to show the result above the form.
- Use the CSV or PDF buttons to export the results table.
FAQs
1. What is theoretical probability?
Theoretical probability measures how likely an event is when every outcome in the sample space is equally likely. It uses favorable outcomes divided by total outcomes.
2. When should I use this calculator?
Use it when you know the full sample space before testing. Common examples include cards, dice, coins, color selections, and simple combinatorial events.
3. What is the complement of an event?
The complement is the probability that the event does not happen. It is found by subtracting the event probability from one.
4. Why does the calculator show simplified fractions?
Simplified fractions make probability easier to interpret and compare. A ratio like 18/36 becomes 1/2, which is clearer and mathematically equivalent.
5. What do repeated trials mean here?
Repeated trials assume the same event is tested independently several times with the same probability each time. That lets the calculator apply binomial formulas.
6. What is the difference between at least and exactly?
Exactly means one precise success count, such as exactly two successes. At least includes that count and every larger count up to the trial limit.
7. Can I use this for experimental probability?
Not directly. Experimental probability comes from observed results after trials. This page focuses on theoretical probability based on known possible outcomes.
8. Why are odds different from probability?
Probability compares favorable outcomes to all outcomes. Odds compare favorable outcomes to unfavorable outcomes, so they describe likelihood in a different ratio format.