Theoretical probability needs equally likely outcomes. For a paper cup that can land up, down or on its side, we cannot count our way to an answer. We can run an experiment.
Idea clave
$$\text{experimental probability} = \frac{\text{number of times the event happened}}{\text{number of trials}}$$ It changes from experiment to experiment. With many trials it tends to get close to the true probability.
Por qué funciona
Each trial is a small piece of evidence. With only a few trials, luck can push the fraction far from the truth. With many trials, the lucky streaks in both directions tend to balance out, so the fraction settles down. This is called the law of large numbers.
Worked examples
Ejemplo resuelto: A paper cup
A paper cup is tossed 40 times and lands on its side 12 times. What is the experimental probability of landing on its side?
- $\frac{12}{40} = \frac{3}{10} = 0.3$.
Respuesta: $\frac{3}{10}$
Ejemplo resuelto: Making a prediction
A player made 18 of her last 25 free throws. About how many of her next 100 would you expect her to make?
- Experimental probability: $\frac{18}{25}$.
- $\frac{18}{25} \times 100 = 72$.
Respuesta: About 72
Ejemplo resuelto: Expected and actual
A fair number cube is rolled 60 times. How many 6s do you expect? If it shows 13 sixes, is the cube unfair?
- Expected: $\frac{1}{6} \times 60 = 10$.
- 13 is close to 10. Differences like this happen by chance all the time, so 13 sixes is no reason to call the cube unfair.
Respuesta: About 10; 13 is still reasonable
Error común
Expecting exactly ten 6s in 60 rolls. Ten is what you expect on average; any one experiment will usually be a little above or below. Over thousands of rolls the fraction of 6s gets close to $\frac{1}{6}$, but the exact count still varies.