Cambridge Primary Checkpoint · Mathematics 0096 · Stage 6 · Statistics and Probability

Probability

Describe and compare chances with probability words and proportions, decide when events are mutually exclusive, and carry out experiments with small and large numbers of trials to estimate and predict.

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In the curriculum

Use the language of probability and proportion (impossible, unlikely, even chance, likely, certain, “3 out of 10”, “1 in 4 chance”, “more likely than”) to describe and compare outcomes (6Sp.01)
Decide whether two events can happen at the same time or not, and know that events that cannot happen at the same time are called mutually exclusive (6Sp.02)
Recognise that some probabilities can only be found by doing an experiment with a large number of trials (6Sp.03)
Carry out chance experiments and simulations with small and large numbers of trials; predict, analyse and describe how often outcomes happen, using the language of probability (6Sp.04)

Probability is the maths of chance: how likely something is to happen. You use it when you say “It will probably rain” or “I have a 1 in 6 chance of rolling a six”. At Stage 6 you describe chances with words and proportions, decide when events cannot happen together, and carry out experiments to estimate chances.

Key words

Word / phrase	Meaning / use
probability / chance	الاحتمال / الفرصة — how likely something is to happen
outcome	ناتج — one possible result, e.g. rolling a 4
event	حدث — one or more outcomes, e.g. rolling an even number
impossible	مستحيل — cannot happen
unlikely	غير مرجّح — could happen, but less than half the time
even chance	فرصة متساوية (50-50) — as likely to happen as not
likely	مرجّح — will happen more than half the time
certain	مؤكد — must happen
equally likely	متساويان في الاحتمال — have the same chance
mutually exclusive	حدثان متنافيان — cannot happen at the same time
trial	محاولة — doing the experiment once (one throw, one spin)
fair / biased	عادل / متحيّز — every outcome equally likely / some outcomes more likely
simulation	محاكاة — using a dice, coin or spinner to act out a real situation

1. The language of probability

We put events on a probability scale from impossible to certain. Halfway is even chance. Sometimes the scale is numbered from 0 (impossible) to 1 (certain), with ½ for even chance.

Rolling an ordinary 1–6 dice. A: rolling a 7 (impossible). D: rolling a 6 (unlikely: 1 out of 6). B: a fair coin landing heads (even chance). E: rolling a number bigger than 1 (likely: 5 out of 6). C: rolling a number less than 7 (certain).

Using proportion. When all the outcomes are equally likely (a fair dice, a fair coin, a spinner with equal sections, counters of the same size in a bag), count:

chance = number of ways the event can happen out of the total number of equally likely outcomes

A fair spinner with 8 equal sections.

Event	Out of 8	In words
red	4 out of 8 = half	even chance
blue	2 out of 8 = 1 out of 4 (a 1 in 4 chance)	unlikely
green	1 out of 8	unlikely (very unlikely)
not red	4 out of 8 = half	even chance
red or blue	6 out of 8 = ¾	likely
purple	0 out of 8	impossible

Comparing outcomes. Red is more likely than blue (4 sections compared with 2) — in fact twice as likely. Green and yellow are equally likely (1 section each). Blue is less likely than red but more likely than green.

Size matters, not just colour count

Counting only works when the outcomes are equally likely. If a spinner has one huge red section and three small sections, red is not “1 out of 4” — it is much more likely. Look at the size of each section.

2. Events that can and cannot happen together

Think about one trial — one roll of a dice, one spin, one counter taken from a bag. Two events are mutually exclusive if they cannot both happen in that one trial. To decide, list the outcomes for each event and see whether any outcome is in both lists.

One roll of a 1–6 dice	Outcomes	Can both happen?	Mutually exclusive?
rolling a 2 / rolling a 5	{2} and {5}	no — you only get one number	yes
rolling an odd number / rolling a 6	{1, 3, 5} and {6}	no — 6 is not odd	yes
rolling an even number / rolling more than 3	{2, 4, 6} and {4, 5, 6}	yes — 4 or 6	no
rolling an even number / rolling a prime number	{2, 4, 6} and {2, 3, 5}	yes — 2 is both	no
rolling less than 3 / rolling more than 4	{1, 2} and {5, 6}	no	yes

Everyday examples: “Today is Saturday” and “today is Sunday” are mutually exclusive. “It is raining” and “it is cold” can happen at the same time, so they are not mutually exclusive. On the spinner above, landing on red and landing on blue are mutually exclusive — the arrow points to only one section.

بالعربي: يعني إيه mutually exclusive؟

حدثان «متنافيان» = مايحصلوش مع بعض في نفس المحاولة.
مثال: في رمية زهر واحدة، «يطلع 2» و«يطلع 5» متنافيان — مستحيل يطلعوا الاتنين.
لكن «رقم زوجي» و«رقم أكبر من 3» مش متنافيين لأن 4 و6 بيحققوا الاتنين.
الطريقة: اكتب نواتج كل حدث، لو في ناتج مشترك → مش متنافيين.

3. Finding probabilities by experiment

For a fair coin or a fair dice we can work out the chances, because every outcome is equally likely. But many probabilities cannot be worked out this way. We can only estimate them by doing an experiment with a large number of trials (or by looking at lots of past data):

a drawing pin landing point up or point down (the two outcomes are not equally likely);
a paper cup landing on its side, upright or upside down;
a home-made or biased spinner or dice;
a bean seed growing, a footballer scoring a penalty, a bus arriving late.

Small and large numbers of trials. Results change from one experiment to another — that is chance. With only a few trials, the results can be far from what you expect. With many trials, the proportion settles down and gives a more reliable estimate. Here are the results of a class tossing a fair coin and recording the running totals:

Number of throws	10	20	50	100	200	500
Number of heads	7	12	22	53	96	254
Proportion of heads	7 out of 10 = 0.7	0.6	0.44	0.53	0.48	0.508

With 10 throws the proportion of heads was 0.7 — far from half. As the number of throws grows, the proportion gets closer and closer to 0.5.

Using experimental results to predict. If a drawing pin landed point up 60 times in 100 throws, we estimate that it lands point up about 60 out of every 100 times (about 3 out of 5). In 400 throws we would predict about 60 × 4 = 240 point up. It will probably not be exactly 240, because of chance.

4. Carrying out a chance experiment or simulation

Question and prediction. E.g. “Is my spinner fair?” Prediction: each of the 4 colours comes up about the same number of times — about 1 out of 4 spins.
Plan. Decide how many trials. Start small (20 spins) and then do many (200 spins, perhaps by combining results from the whole class).
Record the results in a tally chart and find the frequencies.
Analyse. Compare the frequencies with your prediction. For 200 spins of a fair 4-colour spinner, expect about 200 ÷ 4 = 50 of each colour.
Describe using probability language: “Red came up much more often than expected (82 out of 200), so red seems more likely than the other colours. The spinner may be biased.”

A simulation uses a coin, dice or spinner to act out a real situation. Example: a goalkeeper saves about 1 in every 3 penalties. Roll a dice for each penalty: 1 or 2 = save, 3 to 6 = goal. Simulate 30 penalties and count the saves (you would expect about 10).

Expected frequency

Expected number of times = number of trials × the chance as a proportion. A fair dice rolled 60 times: you expect each number about 60 ÷ 6 = 10 times. Real results vary around this — 8 or 13 are normal results, not proof that the dice is unfair.

Worked examples

Worked example 1 — Probability words [2 marks] An ordinary dice numbered 1 to 6 is rolled once. Write impossible, unlikely, even chance, likely or certain for each event: (a) rolling an even number (b) rolling a 0 (c) rolling a number less than 5 (d) rolling a 3.

Event	Ways out of 6	Word
(a) even number	2, 4, 6 → 3 out of 6 = half	even chance
(b) a 0	0 out of 6	impossible
(c) less than 5	1, 2, 3, 4 → 4 out of 6	likely
(d) a 3	1 out of 6	unlikely

Answer: (a) even chance (b) impossible (c) likely (d) unlikely. 2 marks for all four, 1 mark for two or three correct. Common mistake: “less than 5” — do not count 5 itself.

Worked example 2 — Comparing chances with a bag of counters [2 marks] The bag holds 4 red (R), 6 blue (B) and 2 yellow (Y) counters of the same size. One counter is taken without looking.

12 counters: 4 red, 6 blue, 2 yellow.

Total = 4 + 6 + 2 = 12 counters, all equally likely to be picked.
Blue: 6 out of 12 = half → even chance. Blue is the most likely colour.
Red: 4 out of 12 = 1 out of 3. Yellow: 2 out of 12 = 1 out of 6. Red is twice as likely as yellow.
Green: 0 out of 12 → impossible.

Question: How many yellow counters must be added so that taking a yellow is as likely as taking a red? Answer: yellow needs to equal red: 4 − 2 = 2 more yellow counters.

Worked example 3 — Mutually exclusive [2 marks] A spinner is numbered 1 to 8 in equal sections. It is spun once. Are these events mutually exclusive? Explain. (a) “a multiple of 4” and “an odd number” (b) “a number less than 4” and “a prime number”

(a) multiples of 4: {4, 8}; odd numbers: {1, 3, 5, 7}. No number is in both → mutually exclusive.
(b) less than 4: {1, 2, 3}; primes: {2, 3, 5, 7}. 2 and 3 are in both → not mutually exclusive: the spinner can land on 2, which is both.

Exam technique: an explanation must give evidence — name a number that is in both lists, or say that no number is in both. 1 is not a prime number.

Worked example 4 — Small and large numbers of trials [2 marks] Ahmed threw a paper cup 20 times: it landed on its side 14 times. Mona threw the same cup 300 times: it landed on its side 186 times. (a) Whose results give the better estimate? Why? (b) Use Mona’s results to predict how many times the cup lands on its side in 50 throws.

(a) Mona’s, because she did many more trials (300 compared with 20). The proportion settles down with more trials.
(b) Mona: 186 out of 300. Divide both by 6: 31 out of 50. So predict about 31 times out of 50.

Common mistake: saying Ahmed’s results are “wrong”. They are not wrong — 20 trials is just too few to give a reliable estimate. Also, a probability like this can only be found by experiment: the three ways a cup can land are not equally likely.

Worked example 5 — Is the dice fair? [2 marks] Layla rolled a dice 60 times. Results: 1 → 9, 2 → 11, 3 → 8, 4 → 12, 5 → 10, 6 → 10. (a) How many times would you expect each number for a fair dice? (b) Layla says “the dice is unfair because 4 came up most often”. Do you agree?

(a) 60 ÷ 6 = 10 times each.
(b) All the results are close to 10 (between 8 and 12). Small differences like these are expected because of chance.

Answer: (b) No — the results are close to what we expect from a fair dice. To be more confident, she should roll it many more times (e.g. 600).

Common mistakes

Thinking that a result that happened recently is “due” or “less likely” next time. A fair coin has no memory: each throw is still an even chance.
Counting sections of a spinner that are different sizes as equally likely.
Saying two events are mutually exclusive without checking for a shared outcome (even and prime share 2).
Expecting an experiment to give exactly the predicted numbers — results vary because of chance.
Trusting a small number of trials more than a large number.
Using “unlikely” and “impossible” as if they mean the same thing. Unlikely can still happen.

How Checkpoint asks this

“Draw an arrow on the probability scale to show the chance of …” or “Ring the word that describes …”.
“Shade the spinner so that it is more likely to land on grey than white” — count sections and check.
“Tick the pairs of events that are mutually exclusive.”
“Ali throws a drawing pin 200 times… Estimate how many times it lands point up in 1000 throws.”
“Whose results are more reliable? Explain.” — the answer is about the number of trials.
“Explain why …” — always use probability words and numbers from the question.

قاعدة لازم تحفظها

الاحتمال (لو النواتج متساوية) = عدد الطرق اللي الحدث يحصل بيها من إجمالي النواتج.
كل ما تزود عدد المحاولات في التجربة، النتيجة تبقى أدق وأقرب للحقيقة.
فيه احتمالات مانقدرش نحسبها بالعد (زي الدبوس أو الكوباية الورق) — لازم نعمل تجربة بعدد كبير من المحاولات.

Practice

Remember

Scale: impossible – unlikely – even chance – likely – certain (0 to 1).
With equally likely outcomes: chance = number of ways out of the total number of outcomes. Compare with “more likely”, “less likely”, “equally likely”, “twice as likely”.
Mutually exclusive events cannot happen in the same trial; check for a shared outcome.
Some probabilities can only be estimated by experiment. More trials → more reliable estimate.
Expected frequency = number of trials × proportion; real results vary because of chance.

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Checkpoint 6 Maths: The Test and How to Prepare

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