Hard GCSE Sampling and convergence Questions

Work out the theoretical probability from the equally likely outcomes

Because the spinner has $8$ equal sections, so all $8$ outcomes are equally likely, and $1$ of them is red, the theoretical probability of red is $\frac{1}{8} = 0.125$. Both students are estimating this same value.

Work out Sami's relative frequency

Sami got red $7$ times in $40$ spins, so their estimate of the probability is $0.175$.

Work out Tess's relative frequency

Tess got red $84$ times in $800$ spins, so their estimate is $0.105$.

Work out how far Sami's estimate is from the theory

Sami's estimate is $0.05$ away from the theoretical probability. The difference is taken as a positive amount.

Work out how far Tess's estimate is from the theory

Tess's estimate is $0.02$ away from the theoretical probability.

Compare the two differences

$0.02$ is smaller than $0.05$, so Tess's estimate is the one closer to the theoretical probability.

Put both estimates on the probability scale with the theory

Marking all three values on the same scale shows which estimate has landed nearer the theoretical probability.

Plot the two estimates against the size of the sample

The dashed line is the theoretical probability. The picture shows the same thing the arithmetic does: which estimate has landed nearer the line.

Say which sample was the larger

Tess used $800$ spins, more than the other student. A larger sample gives a more reliable estimate ON AVERAGE - it is the one to bet on before the results are seen.

Say why the larger sample is not guaranteed to win

This is the key point. A larger sample TENDS to be closer, but any single small sample can happen to land very close to the theoretical probability by luck. Only the actual differences can decide which of these two is closer, and that is why they were worked out. Here the larger sample (Tess's $800$ spins) has landed closer, as usually happens - but that was not guaranteed in advance.

Rule out the idea that the estimates are equally close

The two differences are not equal, so the two estimates are not the same distance from the theoretical probability.

Rule out the idea that the theory has to wait for the results

The theoretical probability $\frac{1}{8}$ was worked out from the equally likely outcomes before anything was thrown or spun. It does not need the experiment, and the experiment cannot change it.

Say what neither estimate proves

Neither student has proved anything about the probability. Each has made an estimate of it, and one of those estimates happens to be nearer the true value than the other.

Say how either student could do better

The only reliable way to improve an estimate is to carry out more trials, or to pool the two sets of results into one larger sample.

State the answer

Tess's estimate is closer: Tess's relative frequency $0.105$ differs from the theoretical probability $0.125$ by $0.02$, and Sami's differs by $0.05$, so Tess's estimate is closer.

Work out the theoretical probability from the equally likely outcomes

Because a fair coin has $2$ equally likely outcomes, heads and tails, and $1$ of them is heads, the theoretical probability of heads is $\frac{1}{2} = 0.5$. Both students are estimating this same value.

Work out Quinn's relative frequency

Quinn got heads $32$ times in $50$ throws, so their estimate of the probability is $0.64$.

Work out Rosa's relative frequency

Rosa got heads $1024$ times in $2000$ throws, so their estimate is $0.512$.

Work out how far Quinn's estimate is from the theory

Quinn's estimate is $0.14$ away from the theoretical probability. The difference is taken as a positive amount.

Work out how far Rosa's estimate is from the theory

Rosa's estimate is $0.012$ away from the theoretical probability.

Compare the two differences

$0.012$ is smaller than $0.14$, so Rosa's estimate is the one closer to the theoretical probability.

Put both estimates on the probability scale with the theory

Marking all three values on the same scale shows which estimate has landed nearer the theoretical probability.

Plot the two estimates against the size of the sample

The dashed line is the theoretical probability. The picture shows the same thing the arithmetic does: which estimate has landed nearer the line.

Say which sample was the larger

Rosa used $2000$ throws, more than the other student. A larger sample gives a more reliable estimate ON AVERAGE - it is the one to bet on before the results are seen.

Say why the larger sample is not guaranteed to win

This is the key point. A larger sample TENDS to be closer, but any single small sample can happen to land very close to the theoretical probability by luck. Only the actual differences can decide which of these two is closer, and that is why they were worked out. Here the larger sample (Rosa's $2000$ throws) has landed closer, as usually happens - but that was not guaranteed in advance.

Rule out the idea that the estimates are equally close

The two differences are not equal, so the two estimates are not the same distance from the theoretical probability.

Rule out the idea that the theory has to wait for the results

The theoretical probability $\frac{1}{2}$ was worked out from the equally likely outcomes before anything was thrown or spun. It does not need the experiment, and the experiment cannot change it.

Say what neither estimate proves

Neither student has proved anything about the probability. Each has made an estimate of it, and one of those estimates happens to be nearer the true value than the other.

Say how either student could do better

The only reliable way to improve an estimate is to carry out more trials, or to pool the two sets of results into one larger sample.

State the answer

Rosa's estimate is closer: Rosa's relative frequency $0.512$ differs from the theoretical probability $0.5$ by $0.012$, and Quinn's differs by $0.14$, so Rosa's estimate is closer.

Work out the theoretical probability from the equally likely outcomes

Because the spinner has $4$ equal sections, so all $4$ outcomes are equally likely, and $1$ of them is blue, the theoretical probability of blue is $\frac{1}{4} = 0.25$. Both students are estimating this same value.

Work out Omar's relative frequency

Omar got blue $18$ times in $60$ spins, so their estimate of the probability is $0.3$.

Work out Priya's relative frequency

Priya got blue $168$ times in $600$ spins, so their estimate is $0.28$.

Work out how far Omar's estimate is from the theory

Omar's estimate is $0.05$ away from the theoretical probability. The difference is taken as a positive amount.

Work out how far Priya's estimate is from the theory

Priya's estimate is $0.03$ away from the theoretical probability.

Compare the two differences

$0.03$ is smaller than $0.05$, so Priya's estimate is the one closer to the theoretical probability.

Put both estimates on the probability scale with the theory

Marking all three values on the same scale shows which estimate has landed nearer the theoretical probability.

Plot the two estimates against the size of the sample

The dashed line is the theoretical probability. The picture shows the same thing the arithmetic does: which estimate has landed nearer the line.

Say which sample was the larger

Priya used $600$ spins, more than the other student. A larger sample gives a more reliable estimate ON AVERAGE - it is the one to bet on before the results are seen.

Say why the larger sample is not guaranteed to win

This is the key point. A larger sample TENDS to be closer, but any single small sample can happen to land very close to the theoretical probability by luck. Only the actual differences can decide which of these two is closer, and that is why they were worked out. Here the larger sample (Priya's $600$ spins) has landed closer, as usually happens - but that was not guaranteed in advance.

Rule out the idea that the estimates are equally close

The two differences are not equal, so the two estimates are not the same distance from the theoretical probability.

Rule out the idea that the theory has to wait for the results

The theoretical probability $\frac{1}{4}$ was worked out from the equally likely outcomes before anything was thrown or spun. It does not need the experiment, and the experiment cannot change it.

Say what neither estimate proves

Neither student has proved anything about the probability. Each has made an estimate of it, and one of those estimates happens to be nearer the true value than the other.

Say how either student could do better

The only reliable way to improve an estimate is to carry out more trials, or to pool the two sets of results into one larger sample.

State the answer

Priya's estimate is closer: Priya's relative frequency $0.28$ differs from the theoretical probability $0.25$ by $0.03$, and Omar's differs by $0.05$, so Priya's estimate is closer.

Read what the table is recording

Each pair is a running total: the number of spins so far, and the number of red in all of those spins.

Work out the theoretical probability from the equally likely outcomes

Because the spinner has $4$ equal sections, so all $4$ outcomes are equally likely, and $3$ of them are red, the theoretical probability of red is $\frac{3}{4} = 0.75$. It is fixed, and no experiment can change it.

Work out the relative frequency at each stage

Each stage gives an estimate of the same probability, from a bigger sample than the stage before it.

Work out the difference at each stage

Each difference is $|\text{relative frequency} - 0.75|$, taken as a positive amount.

Compare the differences

In these results the difference gets smaller at every stage. The estimate is converging on the theoretical probability as the sample grows.

Plot the relative frequency against the size of the sample

The graph shows exactly what the numbers show: wild swings from the smallest sample, then a settling towards the dashed theoretical line.

Check the last estimate against the theory

Even after $2000$ spins the relative frequency is $0.003$ away from $0.75$ - close, but not equal.

Say why the small samples are so far out

With only $20$ spins, one extra success changes the relative frequency by $\frac{1}{20}$. With $2000$ spins it changes it by only $\frac{1}{2000}$, which is why the big samples barely move.

Rule out the claim that the last estimate is exactly the theory

The two numbers are different, so any statement that they are equal is false on the arithmetic alone - and even if they had matched, that would not PROVE the object is fair.

Rule out the claim that a larger sample is guaranteed to be closer

A larger sample TENDS to give a closer estimate, and in this table it does at every stage. But that is a fact about these results, not a guarantee: a bigger sample can land further out by chance.

Rule out the claim that the results show bias

The differences are getting smaller, not bigger, so the results are behaving exactly as a fair spinner should. There is no evidence of bias here.

Say what the table can never show

However far the table were extended, the relative frequency would only ever estimate the probability. It cannot prove the theoretical value.

Check every relative frequency is a possible probability

Successes can never exceed trials, so every relative frequency in the table lies on the $0$ to $1$ scale.

Weigh up the five statements

Only one statement survives the arithmetic: the differences really do shrink at every stage, and they are the numbers worked out above.

State the answer

The statement best supported by these results is: As the sample grows the relative frequency of red gets closer to the theoretical probability $0.75$ at every stage of this table: the differences are $0.15$, $0.07$, $0.02$ and $0.003$.

Read what the table is recording

Each pair is a running total: the number of spins so far, and the number of blue in all of those spins.

Work out the theoretical probability from the equally likely outcomes

Because the spinner has $10$ equal sections, so all $10$ outcomes are equally likely, and $4$ of them are blue, the theoretical probability of blue is $\frac{2}{5} = 0.4$. It is fixed, and no experiment can change it.

Work out the relative frequency at each stage

Each stage gives an estimate of the same probability, from a bigger sample than the stage before it.

Work out the difference at each stage

Each difference is $|\text{relative frequency} - 0.4|$, taken as a positive amount.

Compare the differences

In these results the difference gets smaller at every stage. The estimate is converging on the theoretical probability as the sample grows.

Plot the relative frequency against the size of the sample

The graph shows exactly what the numbers show: wild swings from the smallest sample, then a settling towards the dashed theoretical line.

Check the last estimate against the theory

Even after $2500$ spins the relative frequency is $0.0028$ away from $0.4$ - close, but not equal.

Say why the small samples are so far out

With only $20$ spins, one extra success changes the relative frequency by $\frac{1}{20}$. With $2500$ spins it changes it by only $\frac{1}{2500}$, which is why the big samples barely move.

Rule out the claim that the last estimate is exactly the theory

The two numbers are different, so any statement that they are equal is false on the arithmetic alone - and even if they had matched, that would not PROVE the object is fair.

Rule out the claim that a larger sample is guaranteed to be closer

A larger sample TENDS to give a closer estimate, and in this table it does at every stage. But that is a fact about these results, not a guarantee: a bigger sample can land further out by chance.

Rule out the claim that the results show bias

The differences are getting smaller, not bigger, so the results are behaving exactly as a fair spinner should. There is no evidence of bias here.

Say what the table can never show

However far the table were extended, the relative frequency would only ever estimate the probability. It cannot prove the theoretical value.

Check every relative frequency is a possible probability

Successes can never exceed trials, so every relative frequency in the table lies on the $0$ to $1$ scale.

Weigh up the five statements

Only one statement survives the arithmetic: the differences really do shrink at every stage, and they are the numbers worked out above.

State the answer

The statement best supported by these results is: As the sample grows the relative frequency of blue gets closer to the theoretical probability $0.4$ at every stage of this table: the differences are $0.15$, $0.07$, $0.028$ and $0.0028$.