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.
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
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.
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 Questions
- State whether each is a probability experiment: (a) measuring the length of a table; (b) rolling a die; (c) choosing a card at random from a deck.
- A spinner is spun 50 times. Red appears 18 times, Blue 17 times, Green 15 times. Find the relative frequency of each colour.
- A biased coin lands heads 65 times in 100 flips. Estimate P(heads) and P(tails).
- Why should you run many trials when estimating probability from an experiment?
- A die is rolled 120 times. Theoretically, how many times would you expect each number to appear? If 3 appears only 8 times, does this mean the die is biased? Explain.