Experimental Probability

Here is a question no amount of clever counting will answer: drop a drawing pin on the table — is it more likely to land point-up or point-down? A die has six matching faces and a coin has two matching sides, so you can count. But a drawing pin has no neat symmetry to lean on. Its two outcomes are simply not equally likely, and there is no way to know which wins just by looking.

So you do the only honest thing: you experiment. Throw the pin many times, keep a tally, and use how often it lands point-up as your best estimate of the probability. That estimate is called the relative frequency:

\text{relative frequency} = \frac{\text{number of times it happened}}{\text{total number of trials}}

The more trials you run, the closer this fraction settles towards the true probability. A few throws tell you little; thousands tell you a lot.

Worked example 1 — reading a frequency table

Here are the results of spinning a coloured spinner 50 times. Each bar is a count; divide by the total to estimate the probability of that colour.

For example, red came up 22 times, so the relative frequency of red is \tfrac{22}{50} = 0.44 — our best estimate of P(\text{red}). Blue's estimate is \tfrac{15}{50} = 0.30, green's is \tfrac{9}{50} = 0.18, and yellow's is \tfrac{4}{50} = 0.08. A neat check: the four estimates add to 0.44 + 0.30 + 0.18 + 0.08 = 1, exactly as probabilities of all the outcomes should.

Worked example 2 — predicting the future

Once you have an estimate, you can turn it around to predict. Rearrange the theorem: the number of times you expect an event is its probability multiplied by the number of trials.

\text{expected frequency} = P(\text{event}) \times \text{number of trials}.

Suppose our drawing-pin experiment gave P(\text{point-up}) \approx 0.6. If we now throw the pin 200 more times, we expect about 0.6 \times 200 = 120 point-ups. Or take the red spinner above at 0.44: in 300 future spins we'd expect roughly 0.44 \times 300 = 132 reds. It's only an expectation — the real count will wobble around it — but it's the best forecast we've got.

Worked example 3 — experiment vs theory for a fair die

For a fair die we already know the theory: P(6) = \tfrac16 \approx 0.167. So here experiment and theory can be compared. Roll the die 60 times and you might get 8 sixes — that's a relative frequency of \tfrac{8}{60} \approx 0.13, close-ish but not exact. Roll it 6000 times and the relative frequency will hug 0.167 far more tightly.

This settling-down is the law of large numbers: as the number of trials grows, the relative frequency closes in on the true probability. Watch a run of coin flips do exactly that — it starts jagged, then calms down towards 0.5:

Experimental probability is an estimate, and small samples are unreliable. Flip a perfectly fair coin just 10 times and you might easily get 7 heads — a relative frequency of 0.7, miles from the true 0.5. This does not mean the coin is biased! Short runs wobble a lot; that's normal.

Only with many trials does the relative frequency settle near the true value. So never declare a coin "unfair" or a pin "loaded" after a handful of throws — you simply don't have enough evidence yet. More trials, more trust.

Almost everyone who deals with uncertainty. Nobody can count the ways it might rain tomorrow, so weather forecasters look back at thousands of past days that looked like today and count how often it rained — "70% chance of rain" is an experimental probability pulled straight from history.

Insurance companies price your policy from the frequency of past accidents; medical trials estimate whether a drug works by how often patients recover; and casinos quietly rely on the law of large numbers — any single gambler might get lucky, but across millions of bets the relative frequencies are rock-steady, so the house always wins in the long run. Experimental probability isn't just a classroom trick; it runs the real world.

See it explained