What's the Difference Between Theoretical and Experimental Probability?

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For example, consider rolling a fair six-sided die and defining event $A$ as rolling a six. Since one of the six equally likely outcomes is favourable,

Suppose the die is then rolled $60$ times and a six appears $8$ times. The experimental probability is

The values are different because experimental results are affected by random variation. As the number of independent trials increases, the experimental probability will generally get closer to the theoretical probability; this is the long-run behaviour of relative frequency.

For instance, if a coin lands heads $18$ times in $30$ tosses, the experimental probability is $\frac{18}{30}=0.6$, while the theoretical probability for a fair coin is $0.5$. This difference alone does not prove the coin is biased; more trials would provide stronger evidence about whether the model is appropriate.

A common misconception is that experimental probability must equal theoretical probability. It does not: theoretical probability describes the model, whereas experimental probability describes the data actually collected. A large difference may result from too few trials, natural randomness, or an unsuitable assumption such as treating a biased die as fair.

Exam technique: Identify whether the question gives a model or observed data. Use favourable outcomes over total equally likely outcomes for theoretical probability, but observed successes over total trials for experimental probability. If asked to compare them, calculate both and explain any difference in context.