📈Experimental Probability

There are two ways to get a probability. You can reason it out from the shape of the thing — a fair die has six equally likely faces, so P(six) = 1/6. That is theoretical probability.

Or you can run the experiment and count. Roll the die 60 times, see a six 13 times, and the experimental probability is 13/60. It is also called the relative frequency:

experimental probability = (times it happened) / (number of trials)

The denominator is the number of trials, not the number of possible outcomes. Dividing 13 by 6 because a die has six faces is the standard error here, and it gives an answer above 1, which is impossible.

Experimental probability is the only kind available when there is nothing to reason about: nobody can compute from first principles how often a hand-carved die lands on a reed, or how often an arrow from this workshop turns out warped.

An experimental probability is an estimate, and estimates wobble. Toss a fair coin 8 times and 6 heads is unremarkable — that is a relative frequency of 0.75, nowhere near 0.5. Toss it 800 times and 600 heads would be astonishing.

The law of large numbers says the relative frequency settles down towards the true probability as the number of trials grows. That is why, given three runs of an experiment, the longest one gives the estimate you should trust — however tidy the short one looks.

A useful way to feel this: in 8 trials, one result changing moves the estimate by 12.5 percentage points. In 800 trials, it moves it by 0.125. The evidence does not get *cleaner* with more trials; it gets *steadier*.

Once you have a probability, theoretical or experimental, you can predict a longer run. The prediction is a straight multiplication:

expected number of successes = probability x number of trials

If a spinner lands on gold with probability 3/8, then in 400 spins expect 3/8 x 400 = 150 gold landings.

The word "expect" is doing careful work. You will almost never see exactly 150. What the calculation gives is the centre of the range the real answer will scatter around — a good bet, not a promise. That is also why a prediction should be reported as "about 150", not "150 exactly".

The same equation runs backwards. If 150 successes came up and the probability is 3/8, then there were 150 / (3/8) = 400 trials.

This is where the topic earns its living. A workshop cannot test every arrow it makes, so it tests a sample and assumes the rest behave the same way.

Test 60 arrows, find 9 warped. Experimental probability of a warped arrow = 9/60 = 0.15 = 15%. For a batch of 400 arrows, expect 0.15 x 400 = 60 warped.

Two habits keep this honest. First, always turn the count into a rate before scaling — 9 means nothing until you know it was 9 out of 60. Second, say "about". The batch could easily have 55 or 66 warped arrows; what the sample gives you is the neighbourhood.

And check the direction: a bigger batch must give a bigger predicted count, so if your answer shrank, you divided where you should have multiplied.