Probability Experiments - Trials and Random Results

In probability, an experiment (sometimes called a trial) is any process that can be repeated and produces a well-defined set of possible results. The key word is random – we do not know in advance exactly which result will occur.

One of the most famous probability experiments in history is Buffon's Needle, devised in 1777 by the French naturalist Georges-Louis Leclerc, Comte de Buffon: drop a needle repeatedly onto a floor marked with evenly spaced parallel lines, and the proportion of drops where the needle crosses a line can be used to estimate the value of π, purely from a physical random experiment. It sounds like a curiosity, but the same principle – running repeated random trials to estimate a value that is hard to calculate directly – is exactly what modern “A/B testing” does at tech companies, showing two different versions of a website to random groups of users and measuring which one performs better.

What Makes Something a Probability Experiment?

It can be repeated under the same conditions.
All possible results are known in advance.
The actual result of any single trial cannot be predicted with certainty.

Common Probability Experiments

Experiment	Description	Possible results
Tossing a coin	Flip one coin once	Heads, Tails
Rolling a die	Roll one six-sided die	1, 2, 3, 4, 5, 6
Drawing a card	Pick one card from a 52-card deck	Any of the 52 cards
Spinning a spinner	Spin a divided wheel	Depends on the number of sections
Picking from a bag	Draw one object without looking	Any object in the bag

Single Trials vs. Repeated Trials

A single trial is one run of the experiment – for example, flipping a coin once.
Repeated trials means running the same experiment many times – for example, flipping the same coin 100 times.
Collecting results from many repeated trials gives experimental (or relative frequency) probability.

Relative Frequency

Relative frequency is how often an outcome occurs compared to the total number of trials.

Relative frequency = Number of times the outcome occurred ÷ Total number of trials

Worked Examples

A die is rolled 60 times. The number 6 appears 12 times. Find the relative frequency of rolling a 6.

Relative frequency = 12 ÷ 60 = 0.2 (or 1/5 or 20%). Theoretical P(6) = 1/6 ≈ 0.167. The experiment ran a reasonable number of trials so the results are close but not identical.

A thumbtack is dropped 200 times. It lands point-up 130 times. Estimate P(point-up).

Relative frequency = 130 ÷ 200 = 0.65. Since theoretical probability is hard to calculate for a thumbtack (unlike a fair die), this experimental value is our best estimate.

Fair vs. Biased

Term	Meaning	Example
Fair	All outcomes are equally likely	A perfectly balanced coin
Biased	Some outcomes are more likely than others	A weighted die that lands on 6 more often

When a device is biased, experimental probability from many trials gives a better estimate of the true probability than theoretical reasoning based on equal likelihood.

Key Takeaways

A probability experiment is a repeatable random process with known possible results.
Relative frequency = frequency of outcome ÷ total trials. It estimates probability from data.
The more trials you run, the closer experimental probability gets to theoretical probability.
Biased experiments require many repeated trials to estimate probabilities reliably.

Practice: Expected Frequency & Fairness

Expected Frequency