GCSE · Maths · AQA · Spec 8300

Relative expected frequency and theoretical probability

Roll a fair dice 60 times and you’d expect 10 sixes. But does “expect” mean you’ll get 10? And what happens to that 10 if you roll 600 times?

A fair dice, rolled again and again

Roll more, expect more sixes — the fraction never moves

the rolls slider: 1/10. Rolls 60. Expected sixes 1/6 × 60 = 10. Expected sixes ÷ rolls 10 ÷ 60 = 1/6

Rolls60Expected sixes1/6 × 60 = 10Expected sixes ÷ rolls10 ÷ 60 = 1/6

The bar is the number of rolls of a fair six-sided dice. Drag it and compare the three readouts: which one changes, and which one refuses to?

Exam line: Expected frequency = probability × number of trials. Divide the expected frequency by the number of trials and you are back at the probability.

What does “expected” actually promise?

Expect 10 sixes. Now what?

A fair dice is rolled 60 times. Someone says: “The expected frequency of sixes is 10.”

Which is closest to what you think that sentence means?

You will get exactly 10 sixes — no more, no fewer.The probability predicts about 10 sixes. The real number could be a bit more or a bit less.It means the probability of rolling a six is 10.If you rolled 120 times instead, the probability of a six would go up.

How sure are you?

I'm sureI think soI'm guessing

You will get exactly 10 sixes — no more, no fewer.

This is a common misconception

Really tempting, because the 10 came out of a calculation. But “expected” doesn’t mean “guaranteed”. Every roll is still down to chance, so 60 real rolls could easily give 8 sixes, or 13. The 10 is what the probability predicts — a sensible guide, not something the dice has to obey.

The probability predicts about 10 sixes. The real number could be a bit more or a bit less.

This matches the evidence

Spot on. 10 is the number the probability points to. A real run of 60 rolls might land on it, or a little above or below it. And notice: 10 out of 60 is 10/60 = 1/6 — the same fraction as the probability of a six.

It means the probability of rolling a six is 10.

This is a common misconception

This one mixes up a count with a probability. 10 is a number of sixes. A probability is a fraction of the rolls, so you have to divide by the number of rolls: 10 ÷ 60 = 1/6. So 10 answers “how many?”, and 1/6 answers “how likely?”.

If you rolled 120 times instead, the probability of a six would go up.

This is a common misconception

More rolls means more sixes expected — 20 in 120 rolls — but the probability hasn’t moved: 20 ÷ 120 is still 1/6. The dice is the same dice, so the chance of a six on any one roll is the same. Only the count grows.

An expected frequency is a prediction of about how many times. Divide it by the number of trials and you’re back at the probability — a number between 0 and 1.

Where does it sit?

Put each one on the probability scale

Each situation gives an expected frequency and a number of trials. Work out expected frequency ÷ trials in your head, then drag each one to where it belongs between 0 (impossible) and 1 (certain).

Seven — Ordinary dice, 60 rolls: expect 0 sevensGreen — Spinner with 5 equal sections, 1 green, 200 spins: expect 40 greensHeads — Fair coin, 50 flips: expect 25 headsBlue — Spinner with 4 equal sections, 3 blue, 40 spins: expect 30 blues1 to 6 — Ordinary dice, 90 rolls: expect 90 numbers less than 7

Now work backwards

Find the line that loses the marks

A spinner is designed so that the expected frequency of red in 80 spins is 60. Find the probability of red, then use it to find the expected frequency of red in 200 spins.

A student’s answer — which line goes wrong?

Probability of red = expected frequency ÷ number of spins
= 80 ÷ 60
= 4/3
Expected reds in 200 spins = 4/3 × 200
= 266.7, so about 267 reds

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

How many times should you expect something to happen — and why dividing by the number of goes always takes you straight back to the probability.

What you need to know

A trial is one go: one roll, one spin, one flip. The number of trials is how many goes there are.
A theoretical probability comes from thinking about the outcomes rather than doing the experiment — like 1/6 for a six on a fair dice, because its 6 faces are equally likely.
Expected frequency = probability × number of trials. It is how many times the probability predicts the outcome will happen: about that many, not exactly.
Relative expected frequency = expected frequency ÷ number of trials. It equals the theoretical probability, whatever the number of trials.
Because it is a probability, it sits on the 0 to 1 scale: 0 is impossible, 0.5 is even chance, 1 is certain.

The big picture

The expected frequency of an outcome is probability × number of trials. Divide it by the number of trials and you get the relative expected frequency, which is exactly the theoretical probability — so it always sits on the 0 to 1 scale.

Key points

1Probability to expected frequency: multiply by the number of trials.

2Expected frequency to probability: divide by the number of trials.

3More trials changes the expected count, never the probability.

4Below 0.5 is unlikely, above 0.5 is likely; 0 and 1 are impossible and certain.

5A probability bigger than 1 means a mistake — usually a division done upside down.

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Worked example

Problem

A fair spinner has 8 equal sections, numbered 1 to 8. It is spun 200 times. Work out the expected frequency of a number greater than 5, and show that the relative expected frequency is the probability.

⚠ Watch out

Dividing the wrong way round: working out number of trials ÷ expected frequency instead of expected frequency ÷ number of trials. If your “probability” comes out bigger than 1, the division is upside down.

🧠

Memory hook

Times to go out, divide to come home: probability × trials gives the count you expect; count ÷ trials brings you home to the probability.

✓

Check yourself

Can you explain why expected sixes ÷ rolls is 1/6 whether you roll a fair dice 60 times or 600 times — and why “expect 10 sixes” is not a promise?

Flashcards

(7)

What is a trial?

One go of an experiment — one roll, one spin, one flip.

How do you work out an expected frequency?

Multiply the probability by the number of trials.

What is the relative expected frequency?

The expected frequency divided by the number of trials. It is always equal to the theoretical probability.

You double the number of trials. What happens to the expected frequency and to the probability?

The expected frequency doubles. The probability — the relative expected frequency — stays exactly the same.

Does “expected frequency of 10” mean it will happen exactly 10 times?

No. It is what the probability predicts — the real count can be a bit more or a bit less.

What do 0, 0.5 and 1 mean on the probability scale?

0 = impossible, 0.5 = even chance, 1 = certain. Below 0.5 is unlikely; above 0.5 is likely.

Why can a relative expected frequency never be more than 1?

You can’t expect an outcome more times than there are trials, so expected frequency ÷ trials is at most 1.

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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Expect 10 sixes. Now what?

Put each one on the probability scale

Find the line that loses the marks

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