GCSE Sampling and convergence Practice Questions

Write down the rule for relative frequency

Relative frequency (experimental probability) is worked out from the results of an experiment: how many times the event happened, divided by how many times the experiment was carried out.

Substitute the results of the experiment

The event landing point up happened $74$ times in $200$ trials, so the relative frequency is $\frac{74}{200}$.

Work out the answer

Dividing $74$ by $200$ gives $0.37$. This is an ESTIMATE of the probability of landing point up, not the probability itself.

Work out the theoretical probability from the equally likely outcomes

Because the spinner has $5$ equal sections, so all $5$ outcomes are equally likely, and $1$ of them is red, the theoretical probability of red is $\frac{1}{5}$. This value is fixed: it does not depend on how many trials are carried out.

Say what a larger sample does

As the number of spins grows, the relative frequency settles down: it varies less and less, and it TENDS towards $\frac{1}{5}$. That is why $4000$ spins is worth far more than $40$ spins.

State the answer

The correct statement is: The relative frequency of red after $4000$ spins will usually be closer to the theoretical probability $\frac{1}{5}$ than the relative frequency after $40$ spins, but it is not certain to be closer.

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 $3$ of them are green, the theoretical probability of green is $\frac{3}{8}$. This value is fixed: it does not depend on how many trials are carried out.

Say what each relative frequency is

Each experiment produces a relative frequency - successes divided by trials - and each one is only an ESTIMATE of $\frac{3}{8}$.

Say what a larger sample does

As the number of spins grows, the relative frequency settles down: it varies less and less, and it TENDS towards $\frac{3}{8}$. That is why $6000$ spins is worth far more than $60$ spins.

Say what a larger sample does NOT do

It does not guarantee anything. A run of $60$ spins could, by luck, land closer to $\frac{3}{8}$ than a run of $6000$ spins. "Usually closer" is true; "certainly closer" is false.

Rule out the idea that the estimate becomes exact

Even after $6000$ spins the relative frequency will almost certainly not be exactly $\frac{3}{8}$. It gets close; it does not land on it.

State the answer

The correct statement is: The relative frequency of green after $6000$ spins will usually be closer to the theoretical probability $\frac{3}{8}$ than the relative frequency after $60$ spins, but it is not certain to be closer.

Read what the table is recording

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

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 $\frac{1}{2} = 0.5$. 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 $|\text{relative frequency} - 0.5|$, 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 $5000$ throws the relative frequency is $0.002$ away from $0.5$ - close, but not equal.

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 the results show bias

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

State the answer

The statement best supported by these results is: As the sample grows the relative frequency of heads gets closer to the theoretical probability $0.5$ at every stage of this table: the differences are $0.2$, $0.09$, $0.028$ and $0.002$.

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.