Understanding Relative Frequency as an Approximation of Probability
Relative frequency is the ratio of the number of times a specific event occurs to the total number of trials conducted. It provides an estimate of the probability of that event when theoretical probability is unknown or difficult to calculate.
For example, if a spinner is spun 50 times and lands on red 15 times, the relative frequency of landing on red is 15 divided by 50, which is 0.3. This suggests the probability of landing on red is approximately 0.3 based on this experiment.
Relative frequency improves as the number of trials increases, approaching the true probability according to the Law of Large Numbers. However, with few trials, the estimate may be less reliable due to randomness.
- Relative frequency = (Number of times event occurs) ÷ (Total number of trials)
- Used to estimate probability from experimental data
- Accuracy improves with more trials
Calculating Expected Frequency Using Probability
Expected frequency predicts how many times an event should occur in a given number of trials or within a population, based on known probability.
The formula is: Expected frequency = Probability of event × Total number of trials.
For example, if the probability of rolling a 4 on a fair six-sided die is 1/6, and the die is rolled 60 times, the expected frequency of rolling a 4 is (1/6) × 60 = 10. This means we expect to see the number 4 about 10 times in 60 rolls, on average.
- Expected frequency = Probability × Number of trials
- Predicts average occurrence of an event
- Example: Expected frequency of rolling a 4 in 60 rolls = (1/6) × 60 = 10
Fairness, Bias, and Randomness in Probability Experiments
Fairness means each outcome has an equal chance of occurring. For example, a fair coin has a 50% chance of landing heads or tails.
Bias occurs when some outcomes are favoured over others, intentionally or unintentionally. For example, a spinner weighted to land more often on blue is biased.
Randomness means outcomes are unpredictable and not influenced by previous results. Even in fair experiments, short-term results can vary due to chance, which is why relative frequency may differ from theoretical probability in small samples.
- Fairness: equal likelihood of outcomes
- Bias: unequal likelihood, favouring some outcomes
- Randomness: unpredictability causing variation in results
Applying Relative and Expected Frequencies in Experiments
To estimate probability using relative frequency, conduct an experiment with many trials, record outcomes, and calculate the ratio for the event of interest.
To calculate expected frequency, use the known or theoretical probability and multiply by the number of trials or population size.
For example, spinning a fair spinner with 4 equal sections 80 times, the probability of landing on any section is 1/4. Expected frequency for one section is (1/4) × 80 = 20. If the actual relative frequency from the experiment is 18, this is close to the expected frequency, showing reasonable agreement.
- Conduct many trials to improve relative frequency estimates
- Use probability × trials to find expected frequency
- Compare relative and expected frequencies to assess fairness and randomness
Limitations and Considerations
Relative frequency is only an approximation and can be affected by the number of trials and experimental conditions.
Expected frequency assumes the probability is known and the trials are independent and fair.
Bias or unfair conditions can cause discrepancies between expected and observed frequencies, so it is important to consider the context and possible sources of error when interpreting results.
- Relative frequency varies with sample size
- Expected frequency assumes known probability and fairness
- Discrepancies may indicate bias or experimental error
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Calculate Relative Frequency | Divide number of times event occurs by total trials | 15 reds in 50 spins → 15/50 = 0.3 | Estimate probability from experimental data |
| Calculate Expected Frequency | Multiply probability by total trials | Probability of 4 on die = 1/6; 60 rolls → (1/6)×60=10 | Predict average occurrences in given trials |
| Assess Fairness and Bias | Check if outcomes are equally likely or if some are favoured | Spinner weighted to land on blue more often is biased | Explain differences between observed and expected results |
| Interpret Randomness | Understand short-term variation in results | Getting 18 reds in 80 spins vs expected 20 | Explain why relative frequency may differ from probability |
Active recall
Quick check
1. If a spinner is spun 40 times and lands on green 12 times, what is the relative frequency of green?
Relative frequency = 12 ÷ 40 = 0.3, so the estimated probability of green is 0.3.
2. A fair six-sided die is rolled 120 times. What is the expected frequency of rolling a 5?
Probability of rolling a 5 = 1/6. Expected frequency = (1/6) × 120 = 20.
3. Explain why the relative frequency might differ from the theoretical probability in an experiment.
Because of randomness and the limited number of trials, the relative frequency may vary from the theoretical probability, especially in small samples.
4. What does it mean if a spinner is biased?
It means some outcomes are more likely than others, so the spinner does not have equal chances for all sections.