Welcome to Relative Frequency and Experimental Probability!
Have you ever flipped a coin three times and gotten "Heads" every single time? Mathematically, we expect Heads to land half the time, so why did that happen? Welcome to the fascinating world of experimental probability and relative frequency!
In this chapter of Handling Data for CCEA GCSE Mathematics (2210), you will discover how we use real-life experiments and surveys to estimate the probability of events. Don't worry if probability has felt confusing in the past — we will break down every idea step by step with clear formulas and practical examples.
1. Key Terms: Getting the Basics Clear
Before jumping into calculations, let's make sure we understand the key vocabulary:
Trial: A single performance or repetition of an experiment (for example, rolling a die once, flipping a coin once, or checking one car at a traffic light).
Theoretical Probability: The calculated likelihood of an event based on known, equally likely mathematical outcomes.
Example: A standard fair six-sided die has theoretical probability \(P(6) = \frac{1}{6}\). A fair coin has \(P(\text{Heads}) = 0.5\).
Relative Frequency (Experimental Probability): The proportion of times an outcome actually happens during an experiment or survey.
Expected Frequency: The number of times we predict an event will occur over a certain number of trials based on probability.
Key Takeaway: Theoretical probability is what should happen in theory. Experimental probability (relative frequency) is what actually happens when we do the test!
2. Calculating Relative Frequency
When you conduct an experiment, you can calculate the relative frequency of an event using a very simple division.
The Relative Frequency Formula
\(\text{Relative Frequency} = \frac{\text{Number of times an outcome occurs (Frequency)}}{\text{Total number of trials}}\)
Memory Trick: Think of the word "FO" — Frequency Over total trials! Always put the count of what you are measuring on top and the total number of attempts on the bottom.
How to Write Your Answer
In CCEA GCSE Mathematics, probabilities and relative frequencies must always be written as:
• Fractions: e.g., \(\frac{3}{10}\) or \(\frac{7}{50}\)
• Decimals: e.g., \(0.3\) or \(0.14\)
• Percentages: e.g., \(30\%\) or \(14\%\)
All probability values must strictly lie between \(0\) and \(1\) (or \(0\%\) and \(100\%\)).
Important Examiner Warning: Never write probability as a ratio like \(1:5\) or a word like "unlikely" unless explicitly asked!
Worked Example 1: Rolling a Die
Question: A student rolls a die \(50\) times. The number \(4\) lands \(12\) times. What is the relative frequency of rolling a \(4\)?
Step 1: Identify the frequency of the event: \(\text{Frequency} = 12\).
Step 2: Identify the total number of trials: \(\text{Total trials} = 50\).
Step 3: Substitute into the formula:
\(\text{Relative Frequency} = \frac{12}{50} = \frac{6}{25} = 0.24\)
Key Takeaway: Relative frequency is found by dividing the observed count by the total number of trials.
3. Sample Size and the Law of Large Numbers
Why do experimental results often look different from theoretical probabilities?
If you flip a coin \(4\) times, you might get \(3\) Heads and \(1\) Tail. The relative frequency of Heads is \(\frac{3}{4} = 0.75\), which is higher than the theoretical probability of \(0.5\). This small variation happens purely due to random chance!
The Effect of Increasing Sample Size
When you increase the number of trials (the sample size):
1. The experiment becomes much more reliable.
2. The relative frequency gets closer and closer to the true theoretical probability.
3. The effect of random chance is reduced.
This principle is known as the Law of Large Numbers.
Choosing the Best Estimate
Examiners frequently ask questions comparing different experiments. Look at this typical scenario:
• Alex tosses a coin \(10\) times and gets \(6\) Heads (\(\text{Relative Frequency} = 0.60\)).
• Beth tosses the same coin \(50\) times and gets \(28\) Heads (\(\text{Relative Frequency} = 0.56\)).
• Charlie tosses the same coin \(500\) times and gets \(255\) Heads (\(\text{Relative Frequency} = 0.51\)).
Which student gives the best estimate for the probability of getting Heads?
Answer: Charlie provides the best estimate because Charlie carried out the largest number of trials (\(500\) trials). A larger sample size gives a more accurate and reliable estimate.
Key Takeaway: Whenever an exam question asks which estimate is the most reliable, always pick the one with the greatest total number of trials!
4. Calculating Expected Frequency
Once you know the probability of an event (either theoretical or experimental), you can predict how many times it will happen in future trials. This prediction is called the Expected Frequency.
The Expected Frequency Formula
\(\text{Expected Frequency} = \text{Probability} \times \text{Total number of trials}\)
Worked Example 2: Using Theoretical Probability
Question: A fair six-sided die is rolled \(300\) times. How many times would you expect to roll a \(5\)?
Step 1: Find the theoretical probability of rolling a \(5\):
\(P(5) = \frac{1}{6}\)
Step 2: Identify the total number of future trials: \(\text{Trials} = 300\).
Step 3: Multiply probability by trials:
\(\text{Expected Frequency} = \frac{1}{6} \times 300 = 50\)
You would expect to roll a \(5\) exactly \(50\) times.
Worked Example 3: Using Experimental Probability
Question: A factory tests \(200\) light bulbs and finds that \(8\) of them are faulty. In a batch of \(5000\) light bulbs, how many are expected to be faulty?
Step 1: Calculate the relative frequency (experimental probability) of a faulty bulb:
\(\text{Relative Frequency} = \frac{8}{200} = 0.04\)
Step 2: Multiply by the new total batch size (\(5000\)):
\(\text{Expected Frequency} = 0.04 \times 5000 = 200\)
We expect \(200\) light bulbs to be faulty.
Key Takeaway: Expected frequency predicts the count of future events: simply multiply the probability by the number of trials.
5. Testing for Bias and Fairness
How do we know if a coin, die, or spinner is fair (unbiased) or biased (unfair)? We compare the experimental results with what theoretical probability predicts!
How to Analyze for Bias:
Step 1: Calculate the theoretical expected frequency for a fair item.
Step 2: Compare the observed experimental results to the expected results.
Step 3: Check the sample size:
• If the sample size is small (e.g., \(5\) or \(10\) trials), small differences are just natural random variation. You cannot claim bias from a tiny sample!
• If the sample size is large (e.g., hundreds of trials) and the results differ substantially from what was expected, the item is likely biased.
Worked Example 4: Is the Spinner Biased?
Question: A four-section spinner labelled \(1, 2, 3, 4\) is spun \(400\) times. The number \(1\) lands \(180\) times. Is the spinner fair?
Step 1: Calculate the expected frequency if it were fair:
If fair, \(P(1) = \frac{1}{4} = 0.25\).
\(\text{Expected Frequency} = 0.25 \times 400 = 100\).
Step 2: Compare the observed count with the expected count:
The observed count is \(180\), which is significantly higher than the expected count of \(100\).
Step 3: State the conclusion:
Because the experiment had a large number of trials (\(400\)) and \(180\) is much higher than the expected \(100\), there is strong evidence that the spinner is biased towards the number \(1\).
Key Takeaway: Only suspect bias when there is a large difference between observed and expected results over a large number of trials.
6. Common Pitfalls to Avoid
Watch out for these classic exam errors identified by CCEA examiners:
1. Inverting the Fraction:
Do not divide the total trials by the frequency. Remember: \(\frac{\text{Frequency}}{\text{Total Trials}}\), not the other way around!
2. Claiming Bias with Tiny Samples:
If someone flips a coin \(4\) times and gets \(4\) Heads, you cannot conclude the coin is biased. You must state that more trials are needed because small samples naturally vary.
3. Picking the "Neatest" Estimate instead of the Largest Sample:
When asked which experiment gives the best estimate of probability, always choose the one with the highest number of trials, not the one whose numbers look easiest or closest to your guess.
4. Premature Rounding:
When calculating expected frequency from experimental data, keep your relative frequency as an exact fraction or unrounded decimal. Rounding too early will lead to an inaccurate final answer.
5. Giving Word Answers Instead of Numbers:
If an exam question asks for relative frequency, always give a numerical value (fraction, decimal, or percentage), never words like "probable" or "unlikely".
7. Quick Summary Checklist
Before your exam, make sure you can answer YES to each of these points:
• Can you calculate relative frequency using \(\frac{\text{Frequency}}{\text{Total trials}}\)?
• Do you remember that larger sample sizes give the most reliable probability estimates?
• Can you calculate expected frequency using \(\text{Probability} \times \text{Total trials}\)?
• Can you determine if an object is fair or biased by comparing observed and expected frequencies over large sample sizes?