Experimental vs Theoretical Probability

Two ways to find a probability

There are two routes to a probability, and good data handlers know both.

• Theoretical probability is worked out by thinking. When all outcomes are equally likely, you count favourable outcomes over total outcomes — no experiment required.
• Experimental probability is worked out by doing. You run trials, record what happens, and use the results.

Theoretical probability = favourable outcomes ÷ total possible outcomes Experimental probability (relative frequency) = number of successes ÷ number of trials

Worked example: a fair die

A fair six-sided die has equally likely outcomes, so theory works perfectly.

• P(rolling a 5) = 1 favourable ÷ 6 outcomes = 1/6 ≈ 0.167.

Now roll it for real. Suppose in 60 rolls you get a 5 eleven times.

• Experimental probability of a 5 = 11 ÷ 60 = 0.183.

The two values are close but not identical, and that is completely normal. Real results scatter around the theoretical value.

Why more trials help

The more trials you run, the closer your experimental probability usually gets to the theoretical one. Watch how the relative frequency of heads settles down as a coin is flipped more times.

Number of flips	Heads counted	Relative frequency
10	7	0.70
50	31	0.62
200	109	0.545
1000	503	0.503

After 10 flips the estimate is wild (0.70), but by 1000 flips it is hugging the theoretical 0.5. This long-run steadying is called the law of large numbers.

Predicting how often something happens

Rearranging the relative frequency formula lets you predict an outcome's frequency:

Expected frequency = probability × number of trials

Example: if you roll a fair die 300 times, the expected number of 2s is (1/6) × 300 = 50. You will rarely get exactly 50, but it tells you what to expect.

Spotting a biased object

This is where experiments beat theory. Theory assumes a die is fair, but what if it is not? A die might be weighted so some faces come up more often. You cannot tell by thinking — you must test it.

Example: a suspicious die is rolled 600 times and lands on 6 a total of 240 times.

• Theory for a fair die predicts (1/6) × 600 = 100 sixes.
• The die actually gave 240, well over double.

The experimental probability is 240/600 = 0.4, not the fair value of 0.167. The sensible conclusion is that the die is biased toward 6. For a biased object, experimental probability is the only way to estimate the true chances.

Activity: test a drawing pin

A drawing pin can land point up or point down, but you cannot predict which is more likely by theory — the shape is uneven. So experiment.

1. Drop a pin 50 times and tally "up" and "down".
2. Work out the relative frequency of each.
3. Combine results with classmates to reach hundreds of drops.
4. Watch your estimate steady as the number of trials grows.

Why this matters

Real life is full of events that are not perfectly fair — weather, sports, manufacturing faults, medical outcomes. Experimental probability lets us estimate chance from evidence, while theoretical probability gives a benchmark to compare against. Together they are the backbone of statistics. Revisit the fundamentals in probability basics, and sharpen the fraction work in introduction to fractions.