Relative Frequency & Expected Outcomes
Relative Frequency & Expected Outcomes
Theoretical vs Experimental Probability
Theoretical probability is calculated from equally likely outcomes (e.g. P(heads) = 1/2 for a fair coin).
Experimental probability (also called relative frequency) is based on actual results from an experiment or trial.
Relative frequency = number of times an event occurs ÷ total number of trials
Worked Example: A spinner is spun 200 times. It lands on red 72 times. Find the relative frequency of red.
- Relative frequency = 72/200 = 0.36
Using Relative Frequency
When outcomes are NOT equally likely (e.g. a biased dice), you cannot use theoretical probability. Instead, use relative frequency as an estimate of probability.
Key principle: The more trials you do, the more reliable the estimate. As the number of trials increases, the relative frequency approaches the true probability. This is called the law of large numbers.
Worked Example: A biased coin is flipped. Results so far:
| Number of flips | Heads | Relative frequency of heads |
|---|---|---|
| 10 | 7 | 0.70 |
| 50 | 32 | 0.64 |
| 100 | 58 | 0.58 |
| 500 | 290 | 0.58 |
| 1000 | 573 | 0.573 |
The relative frequency settles towards about 0.57. The best estimate of P(heads) is 0.573 (from the largest number of trials).
Is the Dice/Coin Fair?
Compare the relative frequency to the theoretical probability for a fair object.
Worked Example: A dice is rolled 300 times and shows a six 68 times.
- Relative frequency of six = 68/300 = 0.227
- Theoretical probability for a fair dice = 1/6 ≈ 0.167
- The relative frequency is noticeably higher than expected — this suggests the dice may be biased towards six
However, 300 rolls may not be enough to be certain. More trials would give more confidence.
Expected Outcomes
Expected frequency = probability × number of trials
This tells you how many times you would expect an event to occur. Actual results may differ.
Worked Example: The probability of winning a game is 0.35. If you play 80 games, how many wins do you expect?
- Expected wins = 0.35 × 80 = 28
Worked Example: A bag has red, blue and green counters. P(red) = 0.4, P(blue) = 0.25. If 200 counters are drawn (with replacement), find the expected number of each colour.
- P(green) = 1 − 0.4 − 0.25 = 0.35
- Expected red = 0.4 × 200 = 80
- Expected blue = 0.25 × 200 = 50
- Expected green = 0.35 × 200 = 70
Using Relative Frequency to Estimate Expected Outcomes
When you have experimental data (no theoretical probability), use relative frequency.
Worked Example: In 150 trials, a machine produces 12 defective items. In a production run of 5000 items, how many defectives are expected?
- Relative frequency of defective = 12/150 = 0.08
- Expected defectives = 0.08 × 5000 = 400
Comparing Expected and Actual Results
If actual results differ significantly from expected results, it may indicate:
- Bias in the experiment
- Too few trials (small sample size)
- The model's assumptions are wrong
Worked Example: A spinner has 5 equal sections numbered 1–5. It is spun 100 times. The expected frequency for each number is 20. Actual results: 1 → 18, 2 → 22, 3 → 19, 4 → 21, 5 → 20.
These are close to 20 each — consistent with a fair spinner. If one number showed 35, that would suggest the spinner is not fair.
Exam Tips
- Relative frequency is an ESTIMATE, not the exact probability — use that word
- The best estimate comes from the LARGEST number of trials
- Expected frequency does not have to be a whole number — do not round unless the context requires it
- "The spinner is biased" requires evidence — quote the relative frequency AND how it differs from the theoretical value
- You cannot prove a dice is fair from experimental data alone — you can only say the results are consistent with being fair