Learn probability & relative frequency
Theoretical probability predicts what should happen when outcomes are equally likely. Relative frequency uses observed results to estimate what is likely to happen in future trials.
You will use observed results to calculate relative frequency and compare it with expected probability.
The idea in plain English
Relative frequency is what actually happened in trials or collected data. It is an experimental estimate of probability.
With more trials, relative frequency often settles closer to the expected probability, but it does not have to match exactly.
How to recognise this kind of question
- The problem reports observed results from trials.
- Words such as out of, recorded, observed, or experimental appear.
- You are asked for relative frequency or experimental probability.
Quick decision: Observed target count divided by total observed trials.
The method
- For theoretical probability, count favorable and possible equally likely outcomes.
- For empirical probability, locate the event's observed frequency.
- Divide the event frequency by the total number of trials.
- Reduce the fraction and compare it with the theoretical prediction when available.
Worked examples
Blue occurred 18 times in 60 spinner trials.
- Reduce by 6.
Answer:
A treatment helped 42 of 60 patients. Find the relative frequency.
- Target count is 42 helped patients.
- Total count is 60 patients.
Answer: 0.70 or 70%
Your turn—with help
A coin landed heads 18 times in 30 tosses. Find the relative frequency of heads.
Need a hint?
Use observed heads over total tosses.
Show the answer and steps
- Convert to 0.60 or 60%.
Answer: 0.60 or 60%
Watch out for this mistake
Incorrect: 18 heads in 30 tosses means P(heads) must always be 60%
Why it fails: Relative frequency describes this experiment, not a permanent guarantee.
Do this instead: Say the observed relative frequency was 60%; a fair coin's theoretical probability remains 50%.
Quick check
Try these without looking. Then open each answer to check your thinking.
7 successes in 20 trials
Need a hint?
Successes over total trials.
Show the answer and steps
- 7 ÷ 20 = 0.35.
Answer: 0.35 or 35%
45 rainy days in 150 days
Need a hint?
Observed rainy days over observed days.
Show the answer and steps
Answer: 0.30 or 30%
12 misses in 80 attempts
Need a hint?
The target event is misses.
Show the answer and steps
- 12 ÷ 80 = 0.15.
Answer: 0.15 or 15%
Common mistakes
- Dividing by the number of categories instead of the number of trials.
- Confusing an observed frequency with a theoretical outcome count.
- Expecting empirical probability to match theory exactly in every sample.
Practice tips
- Add the frequency table before forming the denominator.
- Expect relative frequency to stabilize as the number of trials grows.