The Probability Scale and Relative Frequency

1 The Probability Scale

Every probability is a number from $0$ to $1$ inclusive. Nothing can be more certain than certain, and nothing less likely than impossible.

Probability	Word	Example
$0$	Impossible	Rolling a $7$ on an ordinary die
Close to $0$	Very unlikely	Winning the lottery jackpot
$0.5$	Even chance	A fair coin landing heads
Close to $1$	Very likely	Rolling less than $6$ on a die
$1$	Certain	Rolling a number less than $7$

Never write a probability as a ratio. "$1$ in $4$" is fine in speech, but write it as $\tfrac{1}{4}$, $0.25$ or $25\%$. Writing "$1:4$" means something different (one part to four parts, which is $\tfrac{1}{5}$).

Worked Example 1 — Placing probabilities on the scale

Mark these events on a probability scale: (a) a fair die showing an even number; (b) the sun rising tomorrow; (c) picking a red card from a standard deck; (d) rolling two sixes with two dice.

①(a) $\dfrac{3}{6} = 0.5$ — exactly halfway.

②(b) $1$ — certain (for all practical purposes), at the far right.

③(c) $\dfrac{26}{52} = 0.5$ — also halfway.

④(d) $\dfrac{1}{6} \times \dfrac{1}{6} = \dfrac{1}{36} \approx 0.028$ — very close to $0$.

2 Theoretical Probability

Theoretical probability

$P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$

Theoretical probability is worked out by reasoning about the situation, before any experiment is done. It only works when all outcomes are equally likely.

Worked Example 2 — Cards

A card is drawn at random from a standard pack of $52$. Find (a) $P(\text{a heart})$, (b) $P(\text{a king})$, (c) $P(\text{a picture card})$.

①(a) $13$ hearts out of $52$: $\dfrac{13}{52} = \dfrac{1}{4}$

②(b) $4$ kings out of $52$: $\dfrac{4}{52} = \dfrac{1}{13}$

③(c) Picture cards are J, Q, K in each of $4$ suits: $12$ cards.

④$\dfrac{12}{52} = \dfrac{3}{13}$

3 Relative Frequency (Experimental Probability)

Sometimes you cannot reason out a probability — a drawing pin, a bent coin or a real-world event. Then you must experiment.

Relative frequency

$\text{relative frequency} = \dfrac{\text{number of times the event happened}}{\text{total number of trials}}$

Relative frequency is also called experimental probability or an estimate of probability. It is based on what actually happened, not on theory.

Worked Example 3 — Relative frequency

A drawing pin is dropped $200$ times and lands point up $124$ times. Estimate the probability that it lands point up.

①Relative frequency $= \dfrac{124}{200}$

②$= 0.62$

So $P(\text{point up}) \approx 0.62$. Note the word "estimate" — a different set of $200$ drops would give a slightly different value.

Worked Example 4 — Using relative frequency to predict

In a factory, $12$ out of $400$ items sampled were faulty. Estimate how many faulty items there would be in a batch of $5000$.

①Relative frequency $= \dfrac{12}{400} = 0.03$

②Expected faulty in $5000$ $= 0.03 \times 5000 = 150$ items

4 Theoretical vs Experimental

	Theoretical	Experimental (relative frequency)
How it is found	By reasoning	By carrying out trials
Needs	Equally likely outcomes	Data from an experiment
Exactness	Exact	An estimate
Useful for	Fair dice, coins, cards	Biased objects, real-world events
Improves with	Nothing — it is fixed	More trials

Comparing them is how you detect bias. If an experimental probability stays a long way from the theoretical value even after many trials, the object is probably biased.

Worked Example 5 — Is the die fair?

A die is rolled $600$ times. A six comes up $148$ times. Is the die likely to be fair?

①Theoretical: for a fair die, $P(6) = \dfrac{1}{6} = 0.167$ (3 d.p.).

②Expected number of sixes $= \dfrac{1}{6} \times 600 = 100$.

③Experimental: relative frequency $= \dfrac{148}{600} = 0.247$.

④$148$ is well above the expected $100$, and $600$ trials is a large sample.

The die is probably biased towards six.

Always mention both the size of the difference and the number of trials. A gap of $48$ over $600$ rolls is convincing; the same gap over $20$ rolls would be impossible.

Worked Example 6 — When you cannot conclude bias

A coin is tossed $20$ times and lands heads $13$ times. Can you conclude the coin is biased?

①Relative frequency $= \dfrac{13}{20} = 0.65$, compared with the theoretical $0.5$.

②Expected heads $= 10$, actual $= 13$ — a difference of only $3$.

③With just $20$ trials, this kind of variation happens often by chance.

No — you cannot conclude bias. The sample is too small. Tossing the coin many more times would give a much better estimate.

5 Estimating Probabilities from a Table

Worked Example 7 — A spinner's results

A four-colour spinner is spun $500$ times.

Colour	Red	Blue	Green	Yellow
Frequency	$145$	$118$	$96$	$141$

(a) Estimate $P(\text{green})$. (b) Estimate $P(\text{red or yellow})$. (c) Comment on whether the spinner is fair.

①(a) $\dfrac{96}{500} = 0.192$

②(b) $\dfrac{145 + 141}{500} = \dfrac{286}{500} = 0.572$

③(c) A fair four-section spinner would give about $125$ of each.

④The results $145$, $118$, $96$, $141$ vary quite a lot around $125$ — green in particular is well below.

With $500$ trials this suggests the spinner is probably slightly biased, though the evidence is not overwhelming for blue and yellow.

Check: $145 + 118 + 96 + 141 = 500$ ✓

6 Quick Reference

The scale

$0$ impossible, $0.5$ even chance, $1$ certain. Never outside this range.

Theoretical

$\dfrac{\text{favourable}}{\text{total}}$ — needs equally likely outcomes.

Relative frequency

$\dfrac{\text{times it happened}}{\text{trials}}$ — from real data.

Also called

Experimental probability, or an estimate of probability.

Detecting bias

Compare experimental with theoretical over many trials.

Small samples

Cannot prove bias — the variation could easily be chance.

Predicting

Relative frequency $\times$ new number of trials.

Writing it

Fraction, decimal or percentage — never a ratio.

7 Practice Questions

Question 1

Describe each probability in words: (a) $0$, (b) $0.5$, (c) $0.95$, (d) $0.03$.

▶ Show solution

(a) Impossible (b) An even chance (c) Very likely (d) Very unlikely

Question 2

Explain why a probability of $1.3$ is impossible.

▶ Show solution

A probability is the fraction of outcomes that are favourable, so it can never exceed the total.

The maximum is $1$, meaning the event is certain. A value of $1.3$ would mean "more than certain", which has no meaning.

Question 3

A card is drawn from a standard pack. Find (a) $P(\text{a spade})$, (b) $P(\text{a black ace})$, (c) $P(\text{not a heart})$.

▶ Show solution

(a) $\dfrac{13}{52} = \dfrac{1}{4}$

(b) Two black aces (spades and clubs): $\dfrac{2}{52} = \dfrac{1}{26}$

(c) $1 - \dfrac{1}{4} = \dfrac{3}{4}$

Question 4

A bottle top is flipped $250$ times and lands upright $85$ times. Estimate the probability that it lands upright.

▶ Show solution

Relative frequency $= \dfrac{85}{250} = 0.34$

Question 5

In a quality check, $9$ out of $300$ light bulbs were faulty. Estimate how many would be faulty in a delivery of $12\,000$ bulbs.

▶ Show solution

Relative frequency $= \dfrac{9}{300} = 0.03$

Expected faulty $= 0.03 \times 12\,000 = 360$ bulbs

Question 6

A five-sided spinner is spun $400$ times and lands on section A $130$ times. Compare this with the theoretical probability and comment.

▶ Show solution

Theoretical (if fair): $P(A) = \dfrac{1}{5} = 0.2$, so expect $0.2 \times 400 = 80$.

Experimental: $\dfrac{130}{400} = 0.325$.

$130$ is much greater than the expected $80$, and $400$ trials is a large sample.

The spinner is probably biased towards section A.

Question 7

Explain the difference between theoretical probability and relative frequency, giving one situation where each is the appropriate tool.

▶ Show solution

Theoretical probability is calculated by reasoning about equally likely outcomes, e.g. $P(\text{a } 3) = \tfrac{1}{6}$ on a fair die. It is exact and needs no experiment.

Relative frequency is calculated from the results of trials, e.g. dropping a drawing pin $200$ times. It is only an estimate, but it is the only option when outcomes are not equally likely.

Question 8

A coin is tossed $10$ times and lands heads $7$ times. Josh says "the coin must be biased". Explain why Josh may be wrong.

▶ Show solution

Ten trials is a very small sample. Getting $7$ heads out of $10$ happens quite often with a perfectly fair coin, just by chance.

To decide about bias, Josh would need to toss the coin many more times — several hundred — and see whether the relative frequency stays away from $0.5$.

Question 9

The table shows the results of spinning a three-colour spinner $600$ times.

Colour	Red	Blue	Green
Frequency	$294$	$205$	$?$

(a) Find the missing frequency. (b) Estimate $P(\text{green})$. (c) Is the spinner fair?

▶ Show solution

(a) $600 - 294 - 205 = 101$

(b) $\dfrac{101}{600} = 0.168$ (3 d.p.)

(c) A fair three-colour spinner would give about $200$ of each. The results are $294$, $205$, $101$ — red is far too high and green far too low.

The spinner is biased.

Question 10

A researcher records the relative frequency of a biased coin landing heads after different numbers of tosses.

Tosses	$10$	$50$	$100$	$500$	$2000$
Relative frequency	$0.9$	$0.68$	$0.71$	$0.685$	$0.679$

(a) Which value is the best estimate of $P(\text{heads})$, and why? (b) Estimate the number of heads in $3000$ tosses. (c) Estimate $P(\text{tails})$. (d) Explain the pattern in the table.

▶ Show solution

(a) The value from $2000$ tosses, $0.679$. The larger the number of trials, the more reliable the relative frequency as an estimate of the true probability.

(b) $0.679 \times 3000 = 2037$ heads

(c) $P(\text{tails}) = 1 - 0.679 = 0.321$

(d) The early values jump about a lot ($0.9$ then $0.68$), but as the number of tosses increases the relative frequency settles down towards a value of roughly $0.68$. This is the true probability that the experiment is converging on.

The Probability Scale & Relative Frequency (P3) · GCSE Maths Revision · Created with MathJax