# Theoretical vs. Experimental Probability Theoretical probability is the expected probability of an event.

Experimental probability is what actually happens when we run an experiment.

For example, if we flip a coin once, the theoretical probability that it lands on heads is 50%: Suppose we flip a coin 50 times and it only lands on heads 20 times. The experimental probability is the actual proportion of times the coin landed on heads, which would be 20 / 50 = 40%.

In this case, the experimental probability of landing on heads is 40% and the theoretical probability is 50%. This shows that experimental and theoretical probability do not always match.

Over a large number of trials, the experimental probability and theoretical probability tend to converge. For example, suppose we flip the coin 500 times and it lands on heads 240 times. In this case, the experimental probability is 240 / 500 = 48%, which is much closer to our theoretical probability of 50%.

Let’s run a couple examples using dice to illustrate the differences between theoretical and experimental probability.

## Examples of Theoretical vs. Experimental Probability

Example 1

You roll a die. What is the theoretical probability that it lands on a five? The theoretical probability is 1/6 = 16.6%

Suppose you roll the die 60 times. How many times would you expect it to land on a five?

To find the expected number of times it would land on a five, we simply multiply the theoretical probability by the number of trials:

Expected number of 5’s:  theoretical probability * number of trials  =  1/6 * 60  =  10.

We would expected the die to land on a five 10 times.

Suppose we actually roll the die 60 times and it lands on a five 15 times. What is the experimental probability of the die landing on a five?

The experimental probability is the actual proportion of times the die landed on a five, which would be 15 / 60 = 25%.

Is the experimental or theoretical probability higher in this case?

The theoretical probability was 1/6 = 16.6%.

The experimental probability was 25%.

In this case, the experimental probability was higher.

If you rolled the die 1,000 times, would you expect the theoretical and experimental probability to grow further apart or closer together?

Typically, as we conduct more trials of an experiment, the theoretical and experimental probability converge, so we would expect these two probabilities to grow closer together. In this case, the theoretical probably would remain the same at 16.6%, and we would expect the experimental probability to get closer to 16.6% as we conduct more trials.

Example 2

You roll a die 100 times. The table below shows the number of times it landed on each number: What is the expected number of times you would roll either a one, two, or three?

Theoretical probability: 50% Number of trials: 100

Expected number of times you would roll a one, two, or three  =  50% * 100  = 50

What is the experimental probability of rolling a one, two, or three?

Experimental probability = actual proportion of times it landed on a one, two or three = 50 / 100 = 50% Is the theoretical or experimental probability higher?

In this case, the two probabilities turned out to be equal.