What Is Experimental and Theoretical Probability?
Theoretical probability comes from reasoning about the outcomes: favorable outcomes divided by total equally likely outcomes. Experimental probability comes from counting: the number of times the event happened divided by the number of trials.
Enter both pairs of numbers and the calculator shows the two probabilities, the difference, the number of successes you would expect, and a z score that tells you whether the gap is more than ordinary chance.
With more trials the experimental value usually settles near the theoretical one. That is the law of large numbers.
How to Use the Experimental and Theoretical Probability Calculator
- Enter how many outcomes are favorable and how many there are in total.
- Enter how many times the event happened in your trials.
- Compare the two probabilities and the expected count.
Experimental and Theoretical Probability Formula
Theoretical
Experimental
Experimental and Theoretical Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| Theoretical P | Favorable outcomes ÷ total outcomes | Assumes equally likely outcomes |
| Experimental P | Successes ÷ trials | Also called relative frequency |
| Expected | Theoretical P × trials | |
| z | Difference divided by its standard error | Large values suggest more than chance |
Experimental and Theoretical Probability Worked Example
With the example values loaded in the calculator above, the steps are:
- Theoretical probability = favorable ÷ total = 1 ÷ 6 = 0.16667.
- Experimental probability = successes ÷ trials = 23 ÷ 100 = 0.23.
- Difference = 0.23 − 0.16667 = 0.063333. You would expect 16.67 successes in 100 trials.
- Standard error = √(p(1 − p) ÷ n) = 0.037268, so the difference is 1.7 standard errors.
Result: Theoretical probability = 1/6 = 0.16667; Experimental probability = 23/100 = 0.23.
Worked example: rolling a 3
A fair die has one favorable outcome out of six, so the theoretical probability is 1/6 = 0.1667.
Suppose a 3 came up 23 times in 100 rolls. The experimental probability is 23/100 = 0.23. You would expect about 16.7 threes, so the experiment gave a few more than expected.
Why they differ
Chance causes small differences. With few trials the experimental value can be far from the theoretical one. With many trials it usually gets closer.
Where Experimental and Theoretical Probability Is Used
Math class experiments
Compare dice, coin or spinner results with theory.
Checking fairness
See whether a die or coin behaves as expected.
Law of large numbers
Show how experimental probability changes with more trials.
Common Experimental and Theoretical Probability Mistakes to Avoid
- Use many trials before judging a die or coin.
- A big gap can mean the outcomes are not equally likely.
Experimental and Theoretical Probability Terms and Definitions
- Theoretical probability
- The probability worked out from the possible outcomes.
- Experimental probability
- The relative frequency seen in actual trials.
Experimental and Theoretical Probability FAQs
What is the difference between experimental and theoretical probability?
Theoretical comes from reasoning about the outcomes. Experimental comes from running trials and counting.
How do you calculate experimental probability?
Divide the number of times the event happened by the number of trials.
Why is my experimental result different?
Random variation. The difference tends to shrink as the number of trials grows.
How many trials do I need?
There is no fixed number. More trials give a more reliable estimate.