Theoretical and Experimental Probability

First, what is probability?

Probability is a number that describes how likely an event is to occur. It lies between 0 and 1, or equivalently between 0% and 100%. A probability of 0 means an event is impossible, while a probability of 1 means it is certain. citeturn0search0turn0search1

The two ideas in one minute

1. What is theoretical probability?

Theoretical probability is calculated from the possible outcomes of an experiment. It is especially useful when the outcomes are equally likely. citeturn0search0turn0search2

Example: tossing a fair coin

A fair coin has two equally likely outcomes: Head and Tail. If we want the probability of getting a Head:

P(Head) = 1 ÷ 2 = 1/2 = 0.5 = 50%

We did not need to toss the coin. We simply looked at the possible outcomes. That is why this is theoretical probability.

Another theoretical example: rolling a die

Imagine rolling a fair six-sided die. The possible results are 1, 2, 3, 4, 5 and 6.

Suppose we want the probability of rolling an even number.

Favourable outcomes = 2, 4, 6 → 3 outcomes
Total outcomes = 6

P(even) = 3/6 = 1/2 = 50%

Again, we calculated the answer by thinking about the possible outcomes rather than by actually rolling the die.

2. What is experimental probability?

Experimental probability uses the results of an experiment that has actually been performed. We count how many times the event occurred and compare that with the total number of trials. citeturn0search0turn0search1

Example: actually tossing a coin

Suppose a student tosses a coin 20 times. The results are:

Experimental probability of Heads = 12/20 = 0.6 = 60%.

Notice that the theoretical probability of Heads for a fair coin is 50%, but our experiment gave 60%. That is completely possible. A small number of trials does not have to match the theoretical probability exactly. citeturn0search1

Why are the two answers sometimes different?

This is one of the most important ideas in probability.

If you toss a fair coin 10 times, you might get 7 Heads and 3 Tails. You might even get 8 Heads and 2 Tails. The coin does not have to produce exactly 5 Heads and 5 Tails in every small experiment.

As the number of trials becomes much larger, the experimental relative frequency generally tends to move closer to the theoretical probability. This is described by the law of large numbers. citeturn0search1turn0search3

Theoretical vs experimental probability

A real-life example: bus arrivals

Imagine a student records whether a school bus arrives between 7:45 and 8:00 a.m. over 30 school days.

Suppose the bus arrived during that time on 24 of the 30 days.

This does not mean that the bus is mathematically guaranteed to arrive on time 80% of the time. It means that, in those 30 observed days, the event occurred 24 times out of 30.

This is a good example of when actual data is useful. For many real-world events, there is no simple list of equally likely outcomes from which we can calculate a theoretical probability.

A school example: guessing a multiple-choice question

Suppose a question has 4 answer choices and exactly one is correct. If a student chooses randomly, the theoretical probability of selecting the correct answer is:

P(correct) = 1/4 = 25%

Now imagine the student randomly guesses 40 questions and gets 13 correct. The experimental probability is:

13/40 = 0.325 = 32.5%

The two values are different because the experimental result came from only 40 actual trials. More trials would generally give a relative frequency that moves closer to the theoretical value when the same random conditions are maintained.

One important point: experimental probability is not “wrong”

Students sometimes see a theoretical probability of 50% and an experimental probability of 60% and think that one answer must be wrong. That is not necessarily true.

Theoretical probability describes the expected long-run behaviour of the model, while experimental probability describes what happened in the trials you actually observed. With a limited number of trials, the two values can be different. citeturn0search1turn0search3

How to identify the type in an exam

1. If the question asks you to count possible outcomes, think about theoretical probability.
2. If it gives you the results of an experiment, such as “the event occurred 18 times in 50 trials,” think about experimental probability.
3. For a fair coin, die or other equally likely model, theoretical probability can often be calculated without performing the experiment.
4. For situations such as bus arrivals, machine failures or observed sports performance, actual data may be used to estimate an experimental probability.
5. Always check the denominator: theoretical probability uses total possible outcomes, while experimental probability uses total trials.

Quick comparison with one example

The easiest way to remember everything

Frequently asked questions

Can theoretical and experimental probability be equal?

Yes. An experiment may produce a relative frequency exactly equal to the theoretical probability, although this is not guaranteed in a small number of trials.

Why does experimental probability change?

It depends on the results observed in the trials. If the number of trials changes, the number of successful outcomes may also change, so the calculated relative frequency can change.

Does experimental probability always become exactly equal to theoretical probability?

No. As the number of trials increases, the experimental relative frequency tends to get closer to the theoretical probability under the same conditions, but it does not have to become exactly equal. citeturn0search1

Which probability uses actual data?

Experimental probability uses actual observed data from repeated trials.

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