Experimental Probability & Long-Run Frequency

The formula

To find an experimental probability, count how many times the event occurred and divide by the number of trials. If a die is rolled $60$ times and shows a $6$ fifteen times, the experimental probability of rolling a $6$ is $\dfrac{15}{60} = \dfrac{1}{4}$.

Notice what did not matter: the six faces of the die. Experimental probability ignores the structure of the object and uses only the data. That is the most common trap on this topic — a problem hands you both the setup (faces, sections, marbles) and the actual results, and the right move is to use the results.

Reduce the fraction as you would any probability, and remember it still must land between $0$ and $1$.

Experimental vs. theoretical

Theoretical probability comes from counting equally likely outcomes before you touch anything: a coin has $2$ sides, so $P(\text{heads}) = \dfrac{1}{2}$. Experimental probability comes from running the experiment and recording what happened. Flip that same coin $50$ times and you might get $30$ heads, for an experimental probability of $\dfrac{30}{50} = \dfrac{3}{5}$.

Both numbers are correct — they answer different questions. $\dfrac{1}{2}$ is the prediction; $\dfrac{3}{5}$ is what this particular batch of $50$ flips produced. Small batches of trials bounce around, so the two values usually disagree a little, and that disagreement is normal, not evidence that the coin is unfair.

Long-run frequency: why more trials help

With more trials, experimental probability gets closer to the theoretical probability. Ten flips can easily produce $7$ heads, but $1{,}000$ flips almost never produce $700$ heads — the streaks even out and the fraction settles near $\dfrac{1}{2}$. That settling is what long-run frequency means, and it is why a bigger experiment is always the more trustworthy one.

It is also how probability gets estimated when there is nothing to count. No one can compute the theoretical probability that a basketball player makes a free throw, but $18$ makes out of $24$ attempts gives an experimental probability of $\dfrac{18}{24} = \dfrac{3}{4}$ — and the longer the record, the better that estimate.

Worked examples

Try one yourself

Common questions