International Mathematics (0607) | Relative and expected frequencies

Hello, Future Probability Expert!

Welcome to the exciting world of Relative and Expected Frequencies! If theoretical probability (like P(Heads) = 0.5) tells us what *should* happen, this chapter tells us what *actually* happens when we run an experiment, and how to make predictions based on that.

Don't worry if this sounds complicated—we are just linking the maths you know to real-world results, like checking if a coin is truly "fair" or predicting how many times a team might win based on their past performance. Let's dive in!

Section 1: What is Relative Frequency? (Experimental Probability)

1.1 The Difference Between Theory and Experiment

When you first learned probability (C9.1), you worked with Theoretical Probability. This is based on perfect mathematical scenarios.

• Example: The theoretical probability of rolling a '4' on a fair six-sided die is \(P(4) = \frac{1}{6}\).

However, if you actually roll the die six times, you probably won't get exactly one '4'! This is where Relative Frequency comes in.

1.2 Defining Relative Frequency

Relative Frequency (also known as Experimental Probability) is the estimate of probability based purely on the results of an experiment or observation. It shows the proportion of times an event occurred during the experiment.

Formula for Relative Frequency

The formula is very straightforward:

\[ \text{Relative Frequency} = \frac{\text{Number of times the event occurred}}{\text{Total number of trials (or experiments)}} \]

Remember: The answer should always be between 0 and 1 (or 0% and 100%).

1.3 Step-by-Step Example

Imagine a student spins a biased spinner 200 times and records the outcome:

Outcome (Colour)	Frequency (Number of spins)
Red	45
Blue	105
Green	50

Question: Estimate the probability of spinning Blue using relative frequency.

Step 1: Identify the necessary numbers.
Number of times Blue occurred = 105
Total number of trials = 200

Step 2: Apply the formula.
\[ \text{Relative Frequency (Blue)} = \frac{105}{200} \]

Step 3: Simplify or convert (if needed).
\( \frac{105}{200} = \frac{21}{40} \) (or 0.525 or 52.5%)

Key Takeaway: The Relative Frequency of Blue is 0.525. This is our best estimate of the true probability of spinning Blue, based on the experiment.

1.4 The Law of Large Numbers (Why we experiment)

If you only spin the spinner 5 times, the relative frequency might be wildly inaccurate. However, the more times you repeat an experiment (the larger the number of trials), the closer the Relative Frequency will get to the true Theoretical Probability.

Memory Aid: More trials = Better estimate! If you have to choose between a relative frequency based on 10 trials or 10,000 trials, the 10,000 trials will provide a much more reliable estimate of the probability.

Section 2: Expected Frequency (Making Predictions)

2.1 What is Expected Frequency?

Once we have a known probability (either theoretical or an accurate relative frequency estimate), we can make predictions. The Expected Frequency is the number of times we predict an event will occur if the experiment is repeated a certain number of times.

It is the 'expected value' you get from a population or a set of trials.

Formula for Expected Frequency

To calculate the prediction, you multiply the probability by the total number of attempts:

\[ \text{Expected Frequency} = P(\text{Event}) \times \text{Total number of trials} \]

Did you know? Expected frequency often doesn't result in a whole number. This is totally normal! If you expect 12.5 people to choose vanilla ice cream, it just means 12 or 13 people are most likely to choose it.

2.2 Step-by-Step Example (Using Theoretical Probability)

Question: A fair six-sided die is rolled 300 times. How many times would you expect to roll a number greater than 4?

Step 1: Find the probability of the event.
A number greater than 4 means rolling a 5 or a 6. (2 successful outcomes)
Total outcomes = 6
\(P(\text{Greater than } 4) = \frac{2}{6} = \frac{1}{3}\)

Step 2: Identify the total number of trials.
Total trials = 300

Step 3: Calculate the expected frequency.
\[ \text{Expected Frequency} = P(\text{Event}) \times \text{Total Trials} \] \[ \text{Expected Frequency} = \frac{1}{3} \times 300 = 100 \]

Conclusion: We would expect to roll a number greater than 4 exactly 100 times.

2.3 Example using Relative Frequency to Estimate

Let's go back to the biased spinner from Section 1. We estimated that \(P(\text{Blue}) = 0.525\).

Question: If the spinner is spun 500 more times, what is the expected frequency of spinning Blue?

Step 1: Use the best estimate of probability.
\(P(\text{Blue}) = 0.525\)

Step 2: Identify the total number of trials.
Total trials = 500

Step 3: Calculate the expected frequency.
\[ \text{Expected Frequency} = 0.525 \times 500 = 262.5 \]

Conclusion: We expect Blue to come up 262 or 263 times.

Section 3: Key Vocabulary: Fair, Bias, and Random

These terms help us understand the conditions under which probability experiments are carried out.

3.1 Fair and Bias

These terms usually refer to whether all outcomes in an experiment are equally likely.

• Fair: An object or event is fair if every possible outcome has the same theoretical probability of occurring.
  Example: A standard die is fair if \(P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = \frac{1}{6}\).
• Bias: An object or event has bias (or is biased) if certain outcomes are more likely than others. The theoretical probabilities are not equal.
  Example: A weighted coin might land on heads 80% of the time. This coin is biased towards heads.

We use Relative Frequency results to check for bias. If we flip a coin 1000 times and get 990 heads, we can conclude with high confidence that the coin is biased, even though theoretically it should be fair.

3.2 Random

Random means that the result of any single trial cannot be predicted and is due to chance.

• A random sample is one where every item in the population has an equal chance of being selected.
• A random process (like rolling a die or drawing a card) means the outcome is uncertain until it happens.

Analogy: Even if you know a coin is fair (theoretical probability is 0.5), the outcome of the next toss is still random. Knowing that 5 heads have come up in a row doesn't change the random probability of the 6th toss being tails!