GCSE · Maths · AQA · Spec 8300 · Foundation

Empirical samples and theoretical distributions

Flip a fair coin three times and you might get three heads. Flip it two hundred times and you'll be close to half. Same coin. So what changed?

Watch 201 flips of a fair coin settle down

Number of flips: 101. Relative frequency of heads: 0.49

This is one illustrative run. Drag the point to flip 3, then to flip 145 and flip 149. Flips 1–4 were all heads, and so were flips 146–149. Same streak, very different push.

Theoretical probability: the flat line at 0.5 is the chance of heads on any single flip of a fair coin. The curve is what actually happened: heads so far ÷ flips so far.

Watch out: Even after 201 flips the line reads 0.51, not 0.5, and this coin is perfectly fair. Settling means getting close to the theoretical probability. It doesn't mean landing exactly on it.

What does 'settling down' really mean?

Which idea is closest to yours?

Mia flips a fair coin 1000 times. Early on, she gets five heads in a row.

Which of these is closest to what you think?

After five heads, tails is due. The coin has to catch up so that it all evens out.The number of heads and the number of tails will get closer and closer until the gap between them disappears.Every flip stays 50–50. As the flips pile up, the proportion of heads gets closer to 0.5, but it doesn't have to hit it exactly.With 1000 flips, the relative frequency of heads will come out at exactly 0.5.

How sure are you?

I'm sureI think soI'm guessing

After five heads, tails is due. The coin has to catch up so that it all evens out.

This is a common misconception

Really tempting, but the coin has no memory. Flip six is 50–50, exactly like flip one, and so is every flip after it. The relative frequency does drift back towards 0.5, but not because tails catches up. The five heads simply get outnumbered: they're a huge share of the first few flips, but among 1000 flips they hardly show. Old results never get cancelled. They get diluted.

The number of heads and the number of tails will get closer and closer until the gap between them disappears.

This is a common misconception

It's the proportion that settles, not the gap. Picture 7 heads and 3 tails after 10 flips: a gap of 4, and a relative frequency of 7 ÷ 10 = 0.7. After 1000 flips you might have 512 heads and 488 tails. The gap has grown to 24, yet the relative frequency is 512 ÷ 1000 = 0.512, far closer to 0.5. The counts can drift apart while the proportion closes in.

Every flip stays 50–50. As the flips pile up, the proportion of heads gets closer to 0.5, but it doesn't have to hit it exactly.

This matches the evidence

Yes, that's the whole idea. Each flip stays just as unpredictable as the first one. What becomes predictable is the proportion: with more and more flips it tends to settle closer to the theoretical probability of 0.5, with no guarantee of landing exactly on it.

With 1000 flips, the relative frequency of heads will come out at exactly 0.5.

This is a common misconception

Close, but 'tends towards' doesn't mean 'lands on'. After 1000 fair flips the relative frequency will very probably be near 0.5, something like 0.49 or 0.52, but nothing forces it to be exactly 0.5. The coin on the graph was perfectly fair and still read 0.51 after 201 flips.

Single results stay random for ever. Only the proportion settles, and it settles close to the theoretical probability, not necessarily on it.

Predict, then check

The coin on the graph was fair. What if the experiment isn't?

Spinner A and spinner B each have four equal sections, one of them red. If a spinner is fair, the theoretical probability of red is 1/4 = 0.25. Spinner A is fair. Spinner B has a lump of putty stuck underneath that makes it tend to stop on red. Each is spun 1000 times. What happens to the relative frequency of red?

Your turn to explain it

Whose estimate would you trust?

Kofi and Lena each roll the same fair six-sided dice to estimate the probability of rolling a 6. Kofi rolls it 30 times and gets 9 sixes. Lena rolls it 300 times and gets 52 sixes. Whose results give the more reliable estimate of the probability of rolling a 6? Explain your answer. [3 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

Chooses Lena's results. (1 mark)
Gives the reason: she did more trials (300 rolls compared with 30), a larger sample. (1 mark)
Explains that with more trials the relative frequency tends to be closer to the theoretical probability of 1/6 ≈ 0.17, for example Lena's 52 ÷ 300 ≈ 0.17 while Kofi's 9 ÷ 30 = 0.3 is far off. (1 mark)

Model answer

Lena's. She rolled the dice 300 times, ten times as many as Kofi. The more times you roll, the closer the relative frequency tends to get to the theoretical probability of 1/6 ≈ 0.17. Lena's relative frequency is 52 ÷ 300 ≈ 0.17, while Kofi's is 9 ÷ 30 = 0.3, a long way off. That's the kind of wobble you expect from a small sample.

Name the bigger sample, then say why it's better: more trials means the relative frequency tends to be closer to the theoretical probability.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

relative frequency = number of times it happened ÷ number of trials

Wild in the short run, steady in the long run: the more fair trials you do, the closer this tends to get to the theoretical probability.

What you need to know

Relative frequency = the number of times an outcome happens ÷ the total number of trials.
With only a few trials, the relative frequency can be a long way from the theoretical probability.
As an unbiased (fair) experiment is repeated more and more, the relative frequency tends to settle closer to the theoretical probability.
This happens for every outcome at once: roll a fair dice many times and the relative frequency of each face tends to settle near 1/6. That full set of outcomes and their probabilities is the theoretical probability distribution.
So an estimate from a larger unbiased sample is more reliable than one from a smaller sample.

The big picture

When you repeat a fair experiment, the relative frequency of an outcome (how many times it happened ÷ how many trials) jumps around a lot at first, then settles closer to the theoretical probability as the number of trials grows. So a larger unbiased sample gives a more reliable estimate of a probability. Each single trial stays just as unpredictable, the relative frequency isn't guaranteed to hit the theoretical value exactly, and a biased experiment settles somewhere else.

Key points

1Short run: a handful of results can drag the relative frequency far from the theoretical probability.

2Long run: each new result is a smaller share of the total, so the relative frequency steadies and closes in on the theoretical probability.

3No evening out: every trial stays independent, so a streak never makes the other outcome 'due'.

4Close, not exact: a large fair sample gets near the theoretical probability without being guaranteed to equal it.

5Fair experiments only: more trials steady a biased experiment too, but it settles on its own true probability instead.

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Worked example

Problem

A fair coin is being flipped. (a) After 4 flips there have been 2 heads. The next 3 flips are all heads. Find the relative frequency of heads before and after this streak. (b) After 400 flips there have been 200 heads. The next 3 flips are all heads. Do the same. What does the comparison show?

⚠ Watch out

Thinking a streak makes the other outcome 'due'. After five heads in a row, a fair coin is still 50–50 on the next flip. The relative frequency settles because a streak gets outnumbered by all the flips around it, not because the coin makes up for it.

🧠

Memory hook

Short run, wild. Long run, mild. More trials steady the answer, but they never fix a bias.

✓

Check yourself

A fair spinner has five equal sections, one green. Sam gets green 7 times in 20 spins, then 104 times in 500 spins. Which is the better estimate of the chance of green, and why?

Flashcards

(6)

How do you work out a relative frequency?

Number of times the outcome happened ÷ total number of trials.

What happens to the relative frequency of a fair experiment as the number of trials grows?

It jumps around a lot at first, then tends to settle closer to the theoretical probability.

Why is an estimate from a larger unbiased sample more reliable?

With more trials the relative frequency tends to be closer to the theoretical probability. A small sample can be far off just by chance.

A fair coin lands tails four times running. Is heads now more likely?

No. Each flip is independent, so heads and tails are still 50–50.

Will a very large fair sample give exactly the theoretical probability?

Not necessarily. It tends to get close, with no guarantee of being exactly equal.

Can doing more trials fix a biased spinner?

No. Its results get steadier, but they settle near its own true probability, not the theoretical one for a fair spinner.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Watch 201 flips of a fair coin settle down

Which idea is closest to yours?

Whose estimate would you trust?

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