Experimental probability and expected frequencyIB MYP Maths Standard: Revision notes

Experimental probability is based on results of trials. The relative frequency of an outcome is $\text{relative frequency}=\frac{\text{number of times the outcome happens}}{\text{total number of trials}}.$ Worked example: a spinner lands on red 46 times in 200 spins, so the relative frequency of red is $\frac{46}{200}=0.23$. Use it as an estimate of the probability when you cannot work out the theoretical probability.

If you know (or estimate) a probability, you can predict how many times an outcome will happen: $\text{expected number}=\text{probability}\times\text{number of trials}.$ Example: relative frequency of red $=0.23$, so in 500 spins you expect $0.23\times500=115$ reds. For a fair die rolled 600 times, you expect $\frac{1}{6}\times600=100$ sixes. An expected number is an estimate, not a promise. The actual result is usually close to it but not exactly equal.

The more trials you carry out, the closer the relative frequency is likely to be to the true probability. A result from 10 spins can be very different from the true probability, while a result from 1000 spins is usually much more reliable. Example: relative frequencies $0.42$ (150 flips) and $0.412$ (1000 flips) both estimate the same probability, but the second is more reliable. You can combine results by adding both sets: $\frac{63+412}{150+1000}=0.413$.

Theoretical probability comes from equally likely outcomes, for example $\frac{1}{4}$ for one section of four. Experimental probability comes from trials.

• If they are close, there is no evidence that the object is biased.
• If they stay far apart even after many trials, the object is probably biased. Example: a die gives a relative frequency near $0.2$ for a 6 in 600 rolls and in 6000 rolls. The theoretical value is $\frac{1}{6}\approx0.167$, so the die is probably biased. Always give the numbers and say whether the number of trials was large enough.