Empirical Probability Formula with Examples

Key Points

• The empirical probability formula is $P(E) = \frac{f}{n}$, the number of times an event occurred divided by the number of trials.

• It is computed from observed data, which is why it is also called experimental probability.

• Theoretical probability is derived from the structure of the situation instead, so the two can disagree on any given set of trials.

• The NIST e-Handbook describes a normalised count as the count in a class divided by the total number of observations, which is the same quantity.

• As the number of trials grows, the empirical value tends towards the theoretical one, which is why small samples are unreliable.

What Is the Empirical Probability Formula?

The empirical probability formula estimates a probability from what actually happened:

$$P(E) = \frac{f}{n}$$

Here $f$ is the number of trials in which the event occurred and $n$ is the total number of trials. The NIST e-Handbook defines the equivalent normalised count as the count in a class divided by the total number of observations.

Why Empirical and Theoretical Values Differ

A theoretical probability comes from the structure of the situation. A fair coin has two equally likely faces, so the theoretical probability of heads is exactly $\frac{1}{2}$ before any coin is tossed.

An empirical probability comes from counting. Toss the coin ten times and you might see six heads, giving $\frac{6}{10} = 0.6$, which is not $\frac{1}{2}$ and is not wrong either. It is simply what happened.

The gap narrows as trials accumulate. A thousand tosses will usually give a figure much closer to 0.5, and that convergence is the reason empirical probability is trusted for large samples and distrusted for small ones.

Variable Key

Examples of the Empirical Probability Formula

Example 1

A coin is tossed 40 times and lands heads 22 times. Find the empirical probability of heads.

$$P(\text{heads}) = \frac{22}{40} = 0.55$$

Final answer: 0.55

Example 2

For the same coin, state the probability of heads as 0.5 because the coin is fair.

A fair coin has a theoretical probability of 0.5, so quoting 0.5 looks like the right answer.

But the question asks for the empirical probability, which is computed from the 40 trials actually performed. That gives 0.55, and the difference between the two is the point of the distinction.

$$P(\text{heads}) = \frac{22}{40} = 0.55$$

Final answer: 0.55 empirically, against 0.5 theoretically. Neither figure is wrong; they answer different questions.

Example 3

A machine produces 500 items of which 12 are defective. Find the empirical probability that an item is defective.

$$P(\text{defective}) = \frac{12}{500} = 0.024$$

Final answer: 0.024, or 2.4 per cent.

There is no theoretical model for a machine's defect rate, so empirical probability is the only route here.

Example 4

A die is rolled 60 times and shows a four on 14 occasions. Compare the empirical and theoretical probabilities.

$$\text{empirical} = \frac{14}{60} \approx 0.233, \qquad \text{theoretical} = \frac{1}{6} \approx 0.167$$

Final answer: 0.233 against 0.167.

A gap this size on 60 rolls is unremarkable. It would be far more surprising after 6,000 rolls, which is how unfair dice are detected.

Example 5

In 250 matches a team won 160. Estimate the probability they win their next match, and state the assumption.

$$P(\text{win}) = \frac{160}{250} = 0.64$$

Final answer: 0.64, assuming conditions remain comparable.

That assumption matters. Empirical probability projects past frequencies forward, which only holds while the underlying situation is unchanged.

Empirical, Theoretical and Subjective

The first and last rows are the same arithmetic in different roles. A relative frequency describes a data set you have, while an empirical probability uses that proportion to predict a future trial.

Where Do Students Trip Up on Empirical Probability?

Mistake 1: Giving the theoretical value instead

Where it slips in: The situation is familiar, such as a coin or a die, so the known theoretical probability comes to mind first.

Don't do this: Answer 0.5 for the empirical probability of heads in a set of trials.

The correct way: Count what happened and divide by the number of trials. If the question supplies trial data, it wants the empirical value.

Students who ask whether the question gave them any data before answering get this right consistently. Data in the question is the signal that an empirical answer is wanted, and its absence means the theoretical one.

Mistake 2: Dividing by the number of favourable outcomes

Where it slips in: The two numbers in the question are the frequency and the trial count, and the smaller one can end up underneath.

Don't do this: $\frac{40}{22}$ for 22 heads in 40 tosses.

The correct way: $\frac{22}{40}$. A probability can never exceed 1, so a result above 1 means the fraction is inverted.

Mistake 3: Trusting a small number of trials

Where it slips in: The formula works on any sample size and gives no warning.

Don't do this: Conclude a die is loaded because 3 rolls out of 10 showed a six.

The correct way: Treat small-sample estimates as weak. The swing in short runs is large, and only a sustained gap over many trials is evidence of bias.

Wrapping Up

• The empirical probability formula is $P(E) = \frac{f}{n}$, computed from observed trials.

• It answers a different question from theoretical probability, so a gap between them is expected rather than an error.

• The estimate converges on the theoretical value as trials accumulate, making small samples unreliable.

• Where no model exists, empirical probability is the only way to assign a probability at all.

• To work through probability with a teacher, explore an 8th grade math tutor or math classes online.

Five Minutes of Practice

A spinner is spun 80 times and lands on red 26 times. Find the empirical probability of red. Then compare it with a theoretical $\frac{1}{3}$ for a three-colour spinner, and say what would make the gap convincing evidence of bias.

Want a live Bhanzu trainer to walk through more probability problems? Book a free demo class.

Read More

• Coin Toss Probability Formula

• What Is Probability

• Frequency Formula

• What Is Frequency

• Mean Median Mode Formula