Mathematics (0580) | Relative and expected frequencies

Hello IGCSE Mathematicians! Understanding Relative and Expected Frequencies

Welcome to the exciting world of Experimental Probability! In our previous probability notes, we focused on what *should* happen (like getting a Head 50% of the time when you flip a coin). In this chapter, we look at what *actually* happens when we run experiments, and how we use those real-world results to make predictions.

This skill is crucial not just for exams, but for real life—think about predicting the reliability of a new car model or calculating risk in insurance. Let's master Relative Frequency and Expected Frequency!

Section 1: From Theory to Practice (Experimental Probability)

When studying probability, we differentiate between two main types:

1. Theoretical Probability

This is the probability we calculate based on how we expect a fair experiment to behave. We use logic and formulas:

$$P(\text{Event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$$

Example: The theoretical probability of rolling a '6' on a fair die is \( \frac{1}{6} \).

2. Experimental Probability (Relative Frequency)

This is the probability found by running an experiment many times and recording the results. This probability is based purely on observation.

Example: If you roll a die 60 times and get '6' twelve times, the experimental probability is \(\frac{12}{60}\).

Section 2: Relative Frequency as an Estimate of Probability (C9.2.1 / E9.2.1)

Relative Frequency is another name for Experimental Probability. It tells you how often an event occurred relative to the total number of attempts.

Definition and Calculation

The relative frequency of an event is calculated using the following formula:

$$ \text{Relative Frequency} = \frac{\text{Frequency of Outcome}}{\text{Total Number of Trials (Total Frequency)}} $$

Think of it like a sports statistic: If a footballer takes 50 penalty shots and scores 35 times, their relative frequency of scoring is \(\frac{35}{50} = 0.7\).

Step-by-Step Example: Spinner Experiment

A student spins a four-sided spinner 200 times. The results are recorded below:

Outcome	Frequency
1	45
2	55
3	60
4	40

Question: Estimate the probability of spinning a '3'.

Step 1: Identify the frequency of the required outcome.
Frequency of spinning a '3' = 60.

Step 2: Identify the total number of trials.
Total trials = 200.

Step 3: Calculate the relative frequency.
$$ P(3) \approx \frac{60}{200} = 0.3 $$

The estimated probability of spinning a '3' is 0.3 (or \(\frac{3}{10}\)).

Using Relative Frequency as an Estimate

A crucial idea in this chapter is that Relative Frequency is an estimate of the true probability.

If you run an experiment only a few times (e.g., flipping a coin 10 times), you might get 8 heads. The relative frequency would be 0.8. We know the theoretical probability is 0.5, so 0.8 is a poor estimate.

Key Concept: The Law of Large Numbers

The more times you repeat an experiment (the greater the number of trials), the closer the relative frequency will get to the true theoretical probability.

Imagine flipping that coin 10,000 times. You will almost certainly get a relative frequency very close to 0.5. Therefore, a large number of trials gives a much better estimate.

Section 3: Calculating Expected Frequencies (C9.2.2 / E9.2.2)

Once we know the probability of an event, we can predict how many times it will happen in a future set of trials. This prediction is called the Expected Frequency.

The Expected Frequency Formula

Expected Frequency tells us the average number of times we anticipate an event occurring over a specific number of trials.

$$ \text{Expected Frequency} = P(\text{Event}) \times \text{Total Number of Trials} $$

Don't worry if the answer isn't a whole number! Expected Frequency is a long-term average, so it's perfectly normal to calculate that you expect to see 15.5 rainy days, even though you can't have half a rainy day.

Step-by-Step Example: Predicting Outcomes

The probability that a bus arrives late at a particular stop is \( 0.15 \).

Question: If 80 buses arrive next week, how many are expected to be late?

Step 1: Identify the probability and the total trials.
$$ P(\text{Late}) = 0.15 $$ $$ \text{Total Trials} = 80 $$

Step 2: Apply the Expected Frequency formula.
$$ \text{Expected Frequency} = P(\text{Late}) \times 80 $$ $$ \text{Expected Frequency} = 0.15 \times 80 $$

Step 3: Calculate the result.
$$ 0.15 \times 80 = 12 $$

Conclusion: You expect 12 buses to be late next week.

Using Relative Frequency to Estimate Expected Frequency

Sometimes you are not given the theoretical probability, but you must use results from a previous experiment (the relative frequency) to make a new prediction.

Example: In a factory, 50 items were tested, and 3 were found to be faulty. If the factory produces 600 items tomorrow, how many faulty items are expected?

Step 1: Calculate the Relative Frequency (Estimate of P(Faulty)).
$$ P(\text{Faulty}) \approx \frac{3}{50} = 0.06 $$

Step 2: Calculate the Expected Frequency for the new batch.
$$ \text{Expected Frequency} = P(\text{Faulty}) \times 600 $$ $$ \text{Expected Frequency} = 0.06 \times 600 = 36 $$

We expect 36 items out of the 600 to be faulty.

Section 4: Understanding Fair, Bias, and Random

When working with probability, especially experimental results, it's important to understand the nature of the experiment itself.

1. Fair and Bias (Unbiased vs. Biased)

An object or experiment is Fair (or Unbiased) if all possible outcomes have an equal chance of occurring.

• Example of Fair: A standard, perfectly weighted coin. \(P(H) = P(T) = 0.5\).

An object or experiment is Biased if some outcomes are more likely than others.

• Example of Biased: A loaded die that is weighted so it rolls a '1' more often. \(P(1) > \frac{1}{6}\).

We often use Relative Frequency to test for bias. If you toss a coin 1000 times and get 900 heads (a relative frequency of 0.9), you would strongly suspect the coin is biased!

2. Random

A process is Random if the outcome of any single trial is unpredictable.

• Rolling a die, selecting a marble from a bag without looking, or the arrival time of a bus are all examples of random events.
• In a truly random process, the results of one trial (e.g., rolling a '6') do not affect the results of the next trial.

Key Takeaways and Summary

You have now learned how to connect theoretical mathematics with real-world results!

Relative Frequency:

• It is the result of an experiment: \(\frac{\text{Observed Frequency}}{\text{Total Trials}}\).
• It is used as an estimate for the true probability.
• The more trials, the better the estimate.

Expected Frequency:

• It is a prediction of future outcomes: \( P(\text{Event}) \times \text{Total Trials} \).
• It is a count, not a probability (though it can be non-integer).

Definitions:

• Fair/Unbiased: Equal chance for all outcomes.
• Bias: Unequal chances for outcomes.
• Random: Unpredictable outcome for individual trials.

You're doing great! Keep practicing those calculations, and remember the difference between what we expect and what actually happens.