Probability

Probability questions are short and self-contained, which makes them reliable marks. Two decisions cause nearly all the errors: whether to add or multiply, and whether the totals change between picks.

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1. The basics

P(event) = number of favourable outcomes ÷ total number of outcomes

Probability is always between 0 and 1:

• 0 = impossible
• 1 = certain
• 0.5 = evenly likely

Probability can never be greater than 1 or negative. If your answer is 7/5 or −0.2, something has gone wrong — usually adding when you should have multiplied.

Give answers as fractions, decimals or percentages — all are acceptable unless the question specifies. Fractions do not have to be simplified, but they must be correct.

Example: a bag has 3 red, 5 blue and 2 green counters (10 total).

• P(red) = 3/10
• P(not red) = 7/10
• P(red or green) = 5/10 = 1/2

Always divide by the TOTAL number of outcomes, not by the number of categories. With 10 counters the denominator is 10, not 3.

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2. The complement — the most useful shortcut

P(not A) = 1 − P(A)

All the probabilities of a complete set of outcomes add up to 1.

Example: if P(rain) = 0.6, then P(no rain) = 1 − 0.6 = 0.4.

Subtract from 1, using the right probability. A recorded error was computing 1 − 0.25 when the relevant probability was 0.6. Identify which event you are complementing before subtracting.

Finding a missing probability: if a spinner has P(red) = 0.3, P(blue) = 0.45 and the rest green, then P(green) = 1 − 0.3 − 0.45 = 0.25.

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3. Adding or multiplying — the key decision

Situation	Operation	Words that signal it
One event OR another (mutually exclusive)	ADD	”or”, “either”
One event AND another (both happen)	MULTIPLY	”and”, “both”, “then”

OR means add; AND means multiply. Adding when you should multiply is the most common error in the whole topic — and it usually produces an answer greater than 1, which is your warning sign.

Mutually exclusive events cannot happen together (rolling a 2 and rolling a 5 on one die). For these:

P(A or B) = P(A) + P(B)

Independent events don’t affect each other (two separate coin flips). For these:

P(A and B) = P(A) × P(B)

Example: P(head) = 1/2 and P(rolling a 6) = 1/6.

• P(head and a 6) = 1/2 × 1/6 = 1/12

Multiplying makes probabilities smaller, which makes sense — both things happening is less likely than either alone. If your “and” answer got bigger, you added by mistake.

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4. With and without replacement

This distinction changes the numbers on the second pick.

A bag with 3 red and 5 blue (8 total). Two counters are taken.

With replacement

The first counter goes back, so nothing changes:

• P(red, then red) = 3/8 × 3/8 = 9/64

Without replacement

The first counter is not returned, so both the top and the bottom change:

• P(red, then red) = 3/8 × 2/7 = 6/56 = 3/28

Reduce the numerator AND the denominator. One red is gone, so 3 becomes 2; one counter overall is gone, so 8 becomes 7. Forgetting to adjust the denominator was a specifically recorded error.

Only reduce the numerator if the first item matched. For P(red then blue) without replacement: 3/8 × 5/7 — the blues are untouched, but the total still drops to 7.

“Without replacement” isn’t always stated. Phrases like “takes two counters”, “eats one then another”, or “selects two people” imply it. If items are being consumed or people chosen, they are not going back.

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5. “At least one” — use the complement

P(at least one) = 1 − P(none)

Example: two counters from 3 red and 5 blue, without replacement. Find P(at least one red).

• P(no red) = P(blue, blue) = 5/8 × 4/7 = 20/56 = 5/14
• P(at least one red) = 1 − 5/14 = 9/14

This is far quicker than listing every case. “At least one” would otherwise mean red-blue + blue-red + red-red. The complement collapses it to one calculation.

“At least one” is not “exactly one”. Confusing these was a documented error. At least one includes the case where both happen; exactly one excludes it.

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6. Order matters

When two different outcomes can occur in either sequence, you must count both orders.

Example: P(one red and one blue) from 3 red, 5 blue, without replacement:

• red then blue: 3/8 × 5/7 = 15/56
• blue then red: 5/8 × 3/7 = 15/56
• Total = 30/56 = 15/28

Two different items means two orders — add them. Forgetting the second path halves the answer, and it was flagged by tutors as a recurring loss.

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7. Relative and expected frequency

Relative frequency = number of times it happened ÷ total number of trials

This is experimental probability, used when outcomes are not equally likely — for a biased die or spinner. More trials give a better estimate.

Expected frequency = probability × number of trials

Example: P(red) = 0.3 on a spinner spun 200 times → expected reds = 0.3 × 200 = 60.

An expected frequency is a count, not a probability. The answer is 60, not 0.3 — and it should normally be given as a whole number.

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8. Scope note

Some lessons covered permutations and combinations — notation such as 8C5 and 8P5, and password-counting problems.

These are not on the 0580 syllabus. They belong to Additional Mathematics (0606) and A-level statistics. If you are sitting 0580 only, you do not need nCr or nPr — check your own syllabus document before revising them.

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9. Mistakes that cost marks

Adding when you should multiply (or the reverse).

Giving a probability above 1 without noticing it is impossible.

Dividing by the number of categories instead of the total.

Not reducing the denominator without replacement.

Reducing the numerator when the first pick didn’t match.

Missing that a question implies no replacement.

Treating “at least one” as “exactly one”.

Forgetting the second order for two different outcomes.

Complementing the wrong probability.

Giving a probability when an expected frequency was asked for.

Rounding mid-calculation in multi-stage questions.

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Frequently asked questions

How do I calculate a probability? Favourable outcomes ÷ total outcomes.

What is P(not A)? 1 − P(A) — all probabilities sum to 1.

When do I add and when do I multiply? Add for “or” with mutually exclusive events; multiply for “and” with independent events.

What is the difference between with and without replacement? With replacement, the totals stay the same. Without, both the count of that item and the total decrease by one.

How do I answer “at least one” questions? 1 − P(none).

Is “at least one” the same as “exactly one”? No — “at least one” includes the case where both happen.

Why do I need to consider order? Because “one red and one blue” can happen as red-then-blue or blue-then-red. Add both.

What is relative frequency? Experimental probability: times it happened ÷ number of trials. Used for biased objects.

How do I find expected frequency? Probability × number of trials — the answer is a count.

Do I need combinations (nCr) for 0580? No — that is 0606 material.

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Quick revision checklist

• I know probability is between 0 and 1
• I always divide by the total number of outcomes
• I can use P(not A) = 1 − P(A)
• I can find a missing probability using the total of 1
• I add for “or” and multiply for “and”
• I check my answer isn’t greater than 1
• I can handle with replacement questions
• I reduce both numerator and denominator without replacement
• I spot when a question implies no replacement
• I use 1 − P(none) for “at least one”
• I know “at least one” ≠ “exactly one”
• I count both orders when the two outcomes differ
• I can calculate relative frequency and expected frequency
• I know nCr and nPr are not on 0580

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These notes cover basic probability in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.