Relative and expected frequencies

1. Overview

Relative frequency helps us estimate probabilities based on observed data, especially when we don't know the theoretical probability. Expected frequency allows us to predict how many times an event will occur in a given number of trials, based on its probability. These are essential tools for understanding and predicting real-world events.

Key Definitions

• Relative Frequency: The ratio of the number of times an event occurs to the total number of trials performed. It is an estimate of the probability of the event.
• Expected Frequency: The number of times we predict an event will happen over a specific number of trials, based on its probability.
• Trial: A single observation or one performance of an experiment (e.g., one roll of a die).
• Bias: When the outcomes of an experiment are not equally likely due to an unfair influence (e.g., a weighted coin).

Core Content

3.1 Relative Frequency as an Estimate of Probability

When we don't know the theoretical probability of an event, we use data from experiments or observations. The more trials we conduct, the more reliable the relative frequency becomes as an estimate of the actual probability. This is because with more data, the experimental results tend to get closer to the true probability.

Calculating Relative Frequency:

$\qquad \text{Relative Frequency} = \frac{\text{Frequency of the event}}{\text{Total number of trials}}$

Worked Example 1: A spinner has four colors. It is spun 50 times, and the results are recorded in the table below. Calculate the relative frequency of the spinner landing on "Blue."

Step-by-step working:

1. Identify the frequency of the specific event: The frequency for Blue is 18.
2. Identify the total number of trials: The total number of spins is 50.
3. Apply the formula: $\text{Relative Frequency} = \frac{\text{Frequency of Blue}}{\text{Total number of trials}}$
4. Substitute the values: $\text{Relative Frequency} = \frac{18}{50}$
5. Simplify the fraction: $\text{Relative Frequency} = \frac{9}{25}$
6. Convert to a decimal: $\text{Relative Frequency} = 0.36$ Answer: The relative frequency is $\boxed{0.36}$.

Student Mark Trap: Always ensure you use the total number of trials as the denominator. If the total isn't given, add up all the individual frequencies first.

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3.2 Calculating Expected Frequency

Expected frequency is a prediction. It tells us how many times an event should happen if we know the probability and the number of trials.

Calculating Expected Frequency:

$\qquad \text{Expected Frequency} = \text{Probability} \times \text{Number of trials}$

Worked Example 2: The probability that a seed will germinate is 0.85. If a farmer plants 1,200 seeds, how many seeds would you expect to germinate?

Step-by-step working:

1. Identify the probability ($P$): $P = 0.85$
2. Identify the number of trials ($n$): $n = 1200$
3. Apply the formula: $\text{Expected Frequency} = P \times n$
4. Substitute the values: $\text{Expected Frequency} = 0.85 \times 1200$
5. Calculate: $\text{Expected Frequency} = 1020$ Answer: You would expect $\boxed{1020}$ seeds to germinate.

Worked Example 3 (Theoretical Probability Context): A fair six-sided die is rolled 300 times. How many times would you expect to roll a '4'?

Step-by-step working:

1. Find the probability of the event: For a fair die, $P(\text{rolling a 4}) = \frac{1}{6}$.
2. Identify the number of trials: $n = 300$.
3. Apply the formula: $\text{Expected Frequency} = P(\text{rolling a 4}) \times n$
4. Substitute the values: $\text{Expected Frequency} = \frac{1}{6} \times 300$
5. Calculate: $\text{Expected Frequency} = 50$ Answer: You would expect to roll a '4' exactly $\boxed{50}$ times.

Worked Example 4 — Combining Relative and Expected Frequency A quality control inspector examines a batch of 500 smartphones. She finds that 15 of them have cracked screens. If the factory produces 20,000 smartphones in a month, how many smartphones would you expect to have cracked screens, based on this sample?

Step-by-step working:

1. Calculate the relative frequency of cracked screens in the sample: $\text{Relative Frequency} = \frac{\text{Number of phones with cracked screens}}{\text{Total number of phones inspected}}$
2. Substitute the values: $\text{Relative Frequency} = \frac{15}{500}$
3. Simplify the fraction: $\text{Relative Frequency} = \frac{3}{100} = 0.03$
4. Use the relative frequency to estimate the probability of a cracked screen: $P(\text{cracked screen}) \approx 0.03$
5. Calculate the expected number of cracked screens in the monthly production: $\text{Expected Number} = P(\text{cracked screen}) \times \text{Total number of smartphones produced}$
6. Substitute the values: $\text{Expected Number} = 0.03 \times 20000$
7. Calculate: $\text{Expected Number} = 600$

Answer: You would expect $\boxed{600}$ smartphones to have cracked screens.

Calculator vs Non-Calculator: In non-calculator papers, probabilities are often fractions that cancel out easily with the number of trials (e.g., $\frac{1}{6} \times 300$). In calculator papers, you are more likely to see decimals (e.g., $0.17 \times 450$).

Extended Content (Extended Only)

There is no supplemental content specifically for this sub-topic. All objectives are Core. However, it's important to note that while the calculations are the same for Core and Extended, the complexity of the problems can increase significantly in Extended papers. This often involves multi-step problems where you need to combine relative and expected frequencies with other probability concepts. For example, you might be asked to calculate a relative frequency, then use that to estimate a probability, and then use that probability in a more complex probability calculation (e.g., involving conditional probability or independent events). The key is to break down the problem into smaller, manageable steps and carefully identify what information you are given and what you are trying to find. Always double-check your work and make sure your answers make sense in the context of the problem.

Key Equations

Equation	Meaning of Symbols	Units	Notes
$\text{Relative Frequency} = \frac{f}{n}$	$f = \text{frequency of event}$, $n = \text{total trials}$	None (Ratio/Decimal)	Not on formula sheet
$E = P(A) \times n$	$E = \text{Expected Frequency}$, $P(A) = \text{Probability}$, $n = \text{trials}$	Number of occurrences	Not on formula sheet

Common Mistakes to Avoid

• ❌ Wrong: Thinking the expected frequency must be a whole number. ✓ Right: Expected frequencies can be decimals (e.g., expecting 3.5 heads when flipping a coin 7 times). Do not round unless the question asks for "the number of whole items" or "the nearest whole number".
• ❌ Wrong: Using the frequency instead of the probability to calculate expectation. For example, if 10 out of 50 students are left-handed, and you want to predict how many left-handed students are in a group of 200, you can't just do 10 * 200. ✓ Right: Always convert your "count" into a probability (fraction or decimal) first (10/50 = 0.2), then multiply by the new number of trials (0.2 * 200 = 40).
• ❌ Wrong: Confusing relative frequency with the actual outcome. Just because the relative frequency of rolling a '6' on a die is 0.15 after 100 rolls, doesn't mean you won't roll a '6' on the next roll. ✓ Right: Relative frequency is an estimate based on what has happened; it does not guarantee what will happen in the next trial. It's a long-term average, not a short-term prediction.
• ❌ Wrong: Forgetting to simplify the relative frequency fraction. ✓ Right: Always simplify the fraction to its simplest form, or convert it to a decimal, to make it easier to compare with other probabilities or use in further calculations.

Exam Tips

• Command Words:
  • "Estimate": Usually implies you need to calculate the relative frequency from a table.
  • "Predict" or "Calculate the expected number": Use the Expected Frequency formula.
• Real-world Contexts: Look out for "Quality Control" questions (e.g., testing lightbulbs for defects) or "Fairness" questions (e.g., comparing experimental results to theoretical results to see if a coin is biased). These often require you to calculate a relative frequency and then compare it to a theoretical probability.
• The "More Trials" Rule: If a question asks how to make an estimate of probability more reliable, the answer is always: "Increase the number of trials" or "Conduct the experiment more times." The larger the sample size, the more accurate the relative frequency will be as an estimate of the true probability.
• Probability Range: Remember that relative frequency, like all probabilities, must be between 0 and 1 inclusive. If your answer is greater than 1, you have likely put the fraction upside down. Always check that your answer makes sense in the context of the problem.