Some probabilities can't be calculated theoretically — instead, an experiment (a trial) is repeated many times to see what actually happens. The proportion of times an event occurs is the relative frequency, used as an estimate of the true probability. The more trials carried out, the more reliable that estimate becomes, since a small number of trials can easily give a result that's skewed by chance alone.

Divide the count of the outcome by the total number of trials.

A spinner is spun $80$ times and lands on red $28$ times:

Compare the experimental relative frequency to the theoretical probability expected if the object were fair (every outcome equally likely). A relative frequency a long way from that theoretical value suggests bias.

A coin is flipped $200$ times, landing on heads $134$ times:

1. Relative frequency of heads: $\dfrac{134}{200}=0.67$.
2. Compare with the fair-coin value, $0.5$.
3. Since $0.67$ is considerably higher, the coin appears biased towards heads.

The more trials that are carried out, the closer the relative frequency tends to settle towards the true underlying probability — a spinner tested only $10$ times can easily give a misleading relative frequency purely by chance, whereas the same spinner tested $1000$ times gives a far more trustworthy estimate.

Two students test the same coin: one flips it $20$ times and gets $14$ heads ($0.70$); the other flips it $500$ times and gets $259$ heads ($0.518$). The second student's estimate is more reliable, since a far larger number of trials makes it much less likely that random fluctuation alone explains the result.

Multiply the number of trials by the known (or given) probability of the event — this predicts how many times the event should occur, not a guarantee of exactly how many times it will:

A biased die has $P(\text{six})=0.3$. If rolled $150$ times, the expected number of sixes is:

A biased spinner has $P(\text{blue}) = 0.15$. If the spinner is spun $240$ times, calculate the expected number of times it lands on blue.

Solution

This is a prediction, not a certainty — spinning the actual spinner $240$ times would very likely give a result close to $36$, but rarely exactly $36$.

If an experiment's actual results are noticeably different from the expected frequency, this is evidence (though never absolute proof, since chance is always possible) that the assumed probability was wrong — for example, that a die assumed to be fair might actually be biased.