Empirical Samples

The Law of Large Numbers

Have you ever flipped a coin 10 times and gotten 8 heads, even though you know it should be a 50/50 chance?
This happens because small sample sizes are affected by short-term chance variation, or random fluctuation. However, as the number of trials increases, the relative frequency (also known as experimental probability) gets closer and closer to the theoretical probability.
This convergence principle is known as the Law of Large Numbers. Note that the sum of the relative frequencies for all possible mutually exclusive outcomes in an experiment must always equal $1$ .

\text{Relative Frequency} = \frac{\text{Frequency of event occurring } (f)}{\text{Total number of trials } (n)}

Feature	Small Sample Size (e.g., 10 trials)	Large Sample Size (e.g., 1,000 trials)
Reliability	Unreliable estimate of probability	More reliable estimate of probability
Fluctuation	Highly susceptible to random chance	Smooths out short-term variations
Convergence	Far from theoretical probability	Very close to theoretical probability

Finding the Best Estimate

If three different students do the same coin-flipping experiment, whose result should you trust?
In an exam, you will often need to decide which experiment gives the "best estimate" of a probability. The most reliable estimate always comes from the experiment with the largest number of trials.
When multiple people conduct the same experiment, you do not average their probabilities. Instead, you find the best estimate by combining all their results, totaling the frequencies and the total trial numbers.

Stuart flips a coin 10 times and gets 7 Tails. Maxine flips the same coin 50 times and gets 30 Tails. Calculate the best estimate for the probability of getting Tails.

Step 1: Identify the total number of successful outcomes.

$\text{Total Tails} = 7 + 30 = 37$

Step 2: Identify the total number of trials across all experiments.

$\text{Total Trials} = 10 + 50 = 60$

Step 3: Calculate the best estimate using the combined totals.

$\text{Best Estimate} = \frac{37}{60}$

Avoiding Bias in Experiments

Imagine trying to find the most popular sport in a school by only asking the football team.
For an empirical distribution (the set of observed experimental results) to converge on a theoretical probability distribution, the sample must be an unbiased sample. This means the experiment must be entirely random, every member of the population must have an equal chance of selection, and each trial must be independent.
A sample suffering from bias does not accurately represent the population and does not give every outcome an equal chance of occurring. To ensure fairness, objects must be replaced between trials to keep probabilities constant.
Valid methods to ensure randomness include using a random number generator or picking names from a hat. In stratified sampling, the sample reflects the proportions of different groups in the population.

\text{Sample Size for Stratum} = \frac{\text{Stratum Population}}{\text{Total Population}} \times \text{Total Sample Size}

Using Expected Frequency to Predict Outcomes

Factories test a small batch of products to predict how many faulty items might be in an order of a million.
Once you have calculated a probability, you can use it to estimate future outcomes. The expected frequency is the number of times you estimate an outcome will happen in a given number of trials.
It is calculated by multiplying the probability (theoretical or experimental) by the total number of future trials.

\text{Expected Frequency} = \text{Probability} \times \text{Total number of trials}

A biased 4-sided spinner has a relative frequency of $0.35$ for landing on 'Red'. If the spinner is spun $400$ times, estimate the number of times it will land on Red.

Step 1: State the expected frequency formula.

$\text{Expected Frequency} = \text{Probability} \times \text{Total number of trials}$

Step 2: Substitute the known values into the equation.

$\text{Expected Frequency} = 0.35 \times 400$

Step 3: Calculate the final estimate.

$\text{Expected Frequency} = 140$ times

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Exam Tips

Common Mistake
Students often calculate the relative frequency correctly but forget the final step in a prediction question: you must multiply your relative frequency by the new number of trials to find the expected frequency.
2
When an Edexcel question asks you to 'justify your answer' regarding which estimate is most reliable, the mark scheme strictly requires you to use the phrase 'more trials' or 'larger sample size'.
3
In exams, when asked to choose the 'best estimate' from multiple experiments, always choose the result from the experiment with the largest combined number of trials — never average the individual probabilities.
4
When explaining results, use the term 'estimate' (e.g., 'the estimated probability is 0.4'). Avoid stating 'the probability is 0.4', as experiments only provide estimates, not exact guarantees.
5
Do not round relative frequencies or decimals too early in multi-step problems; use exact fractions or 4 decimal places to avoid accuracy errors when calculating expected frequencies later.

Key Terms(12)

Relative frequency: An estimate of the probability of an event occurring based on data from an experiment or survey.
Experimental probability: Another term for relative frequency; an estimate of probability derived from experimental results rather than theoretical facts.
Theoretical probability: The expected probability calculated based on known facts, logic, or a mathematical model.
Law of Large Numbers: The principle stating that as the number of trials increases, the relative frequency gets closer and closer to the theoretical probability.
Trial: A single performance of an experiment, such as one roll of a die or one flip of a coin.
Empirical distribution: The set of frequencies or probabilities observed in an actual experiment, as opposed to a theoretical model.
Unbiased sample: A representative sample where every member of the population or every possible outcome had an equal chance of being selected.
Population: The entire set of items or people being studied.
Sample: A subset of the population used to collect data to make inferences about the whole.
Bias: A systematic distortion that makes one outcome more likely than it should be theoretically.
Stratified sampling: A method where the sample reflects the exact proportions of different groups (strata) present in the entire population.
Expected frequency: The calculated estimate for the number of successful outcomes in a future set of trials.

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Key Terms(12)

Relative frequency: An estimate of the probability of an event occurring based on data from an experiment or survey.
Experimental probability: Another term for relative frequency; an estimate of probability derived from experimental results rather than theoretical facts.
Theoretical probability: The expected probability calculated based on known facts, logic, or a mathematical model.
Law of Large Numbers: The principle stating that as the number of trials increases, the relative frequency gets closer and closer to the theoretical probability.
Trial: A single performance of an experiment, such as one roll of a die or one flip of a coin.
Empirical distribution: The set of frequencies or probabilities observed in an actual experiment, as opposed to a theoretical model.
Unbiased sample: A representative sample where every member of the population or every possible outcome had an equal chance of being selected.
Population: The entire set of items or people being studied.
Sample: A subset of the population used to collect data to make inferences about the whole.
Bias: A systematic distortion that makes one outcome more likely than it should be theoretically.
Stratified sampling: A method where the sample reflects the exact proportions of different groups (strata) present in the entire population.
Expected frequency: The calculated estimate for the number of successful outcomes in a future set of trials.

Empirical Samples

The Law of Large Numbers

Have you ever flipped a coin 10 times and gotten 8 heads, even though you know it should be a 50/50 chance?
This happens because small sample sizes are affected by short-term chance variation, or random fluctuation. However, as the number of trials increases, the relative frequency (also known as experimental probability) gets closer and closer to the theoretical probability.
This convergence principle is known as the Law of Large Numbers. Note that the sum of the relative frequencies for all possible mutually exclusive outcomes in an experiment must always equal $1$ .

\text{Relative Frequency} = \frac{\text{Frequency of event occurring } (f)}{\text{Total number of trials } (n)}

Feature	Small Sample Size (e.g., 10 trials)	Large Sample Size (e.g., 1,000 trials)
Reliability	Unreliable estimate of probability	More reliable estimate of probability
Fluctuation	Highly susceptible to random chance	Smooths out short-term variations
Convergence	Far from theoretical probability	Very close to theoretical probability

Finding the Best Estimate

If three different students do the same coin-flipping experiment, whose result should you trust?
In an exam, you will often need to decide which experiment gives the "best estimate" of a probability. The most reliable estimate always comes from the experiment with the largest number of trials.
When multiple people conduct the same experiment, you do not average their probabilities. Instead, you find the best estimate by combining all their results, totaling the frequencies and the total trial numbers.

Stuart flips a coin 10 times and gets 7 Tails. Maxine flips the same coin 50 times and gets 30 Tails. Calculate the best estimate for the probability of getting Tails.

Step 1: Identify the total number of successful outcomes.

$\text{Total Tails} = 7 + 30 = 37$

Step 2: Identify the total number of trials across all experiments.

$\text{Total Trials} = 10 + 50 = 60$

Step 3: Calculate the best estimate using the combined totals.

$\text{Best Estimate} = \frac{37}{60}$

Avoiding Bias in Experiments

Imagine trying to find the most popular sport in a school by only asking the football team.
For an empirical distribution (the set of observed experimental results) to converge on a theoretical probability distribution, the sample must be an unbiased sample. This means the experiment must be entirely random, every member of the population must have an equal chance of selection, and each trial must be independent.
A sample suffering from bias does not accurately represent the population and does not give every outcome an equal chance of occurring. To ensure fairness, objects must be replaced between trials to keep probabilities constant.
Valid methods to ensure randomness include using a random number generator or picking names from a hat. In stratified sampling, the sample reflects the proportions of different groups in the population.

\text{Sample Size for Stratum} = \frac{\text{Stratum Population}}{\text{Total Population}} \times \text{Total Sample Size}

Using Expected Frequency to Predict Outcomes

Factories test a small batch of products to predict how many faulty items might be in an order of a million.
Once you have calculated a probability, you can use it to estimate future outcomes. The expected frequency is the number of times you estimate an outcome will happen in a given number of trials.
It is calculated by multiplying the probability (theoretical or experimental) by the total number of future trials.

\text{Expected Frequency} = \text{Probability} \times \text{Total number of trials}

A biased 4-sided spinner has a relative frequency of $0.35$ for landing on 'Red'. If the spinner is spun $400$ times, estimate the number of times it will land on Red.

Step 1: State the expected frequency formula.

$\text{Expected Frequency} = \text{Probability} \times \text{Total number of trials}$

Step 2: Substitute the known values into the equation.

$\text{Expected Frequency} = 0.35 \times 400$

Step 3: Calculate the final estimate.

$\text{Expected Frequency} = 140$ times

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often calculate the relative frequency correctly but forget the final step in a prediction question: you must multiply your relative frequency by the new number of trials to find the expected frequency.
2
When an Edexcel question asks you to 'justify your answer' regarding which estimate is most reliable, the mark scheme strictly requires you to use the phrase 'more trials' or 'larger sample size'.
3
In exams, when asked to choose the 'best estimate' from multiple experiments, always choose the result from the experiment with the largest combined number of trials — never average the individual probabilities.
4
When explaining results, use the term 'estimate' (e.g., 'the estimated probability is 0.4'). Avoid stating 'the probability is 0.4', as experiments only provide estimates, not exact guarantees.
5
Do not round relative frequencies or decimals too early in multi-step problems; use exact fractions or 4 decimal places to avoid accuracy errors when calculating expected frequencies later.

Key Terms(12)

Relative frequency: An estimate of the probability of an event occurring based on data from an experiment or survey.
Experimental probability: Another term for relative frequency; an estimate of probability derived from experimental results rather than theoretical facts.
Theoretical probability: The expected probability calculated based on known facts, logic, or a mathematical model.
Law of Large Numbers: The principle stating that as the number of trials increases, the relative frequency gets closer and closer to the theoretical probability.
Trial: A single performance of an experiment, such as one roll of a die or one flip of a coin.
Empirical distribution: The set of frequencies or probabilities observed in an actual experiment, as opposed to a theoretical model.
Unbiased sample: A representative sample where every member of the population or every possible outcome had an equal chance of being selected.
Population: The entire set of items or people being studied.
Sample: A subset of the population used to collect data to make inferences about the whole.
Bias: A systematic distortion that makes one outcome more likely than it should be theoretically.
Stratified sampling: A method where the sample reflects the exact proportions of different groups (strata) present in the entire population.
Expected frequency: The calculated estimate for the number of successful outcomes in a future set of trials.

Previous Subtopic: Outcome SetsOutcome Sets Next Subtopic: Set EnumerationSet Enumeration

Back to Empirical Samples

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(12)

Relative frequency: An estimate of the probability of an event occurring based on data from an experiment or survey.
Experimental probability: Another term for relative frequency; an estimate of probability derived from experimental results rather than theoretical facts.
Theoretical probability: The expected probability calculated based on known facts, logic, or a mathematical model.
Law of Large Numbers: The principle stating that as the number of trials increases, the relative frequency gets closer and closer to the theoretical probability.
Trial: A single performance of an experiment, such as one roll of a die or one flip of a coin.
Empirical distribution: The set of frequencies or probabilities observed in an actual experiment, as opposed to a theoretical model.
Unbiased sample: A representative sample where every member of the population or every possible outcome had an equal chance of being selected.
Population: The entire set of items or people being studied.
Sample: A subset of the population used to collect data to make inferences about the whole.
Bias: A systematic distortion that makes one outcome more likely than it should be theoretically.
Stratified sampling: A method where the sample reflects the exact proportions of different groups (strata) present in the entire population.
Expected frequency: The calculated estimate for the number of successful outcomes in a future set of trials.