Basic Probability

Say how likely something is in words and on the 0 to 1 scale, work out simple probabilities by counting, use the fact that they add to 1, and work backwards to find how many.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 137 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What Probability Means

Probability is a number that says how likely something is. Every probability is between 0 and 1.

Probability	Means
$0$	impossible
$\frac{1}{2}$	evens: as likely as not
$1$	certain

When every outcome is equally likely, count:

Formula

P = \frac{number of ways it can happen}{total number of outcomes}

Rule

A probability can be a fraction, a decimal or a percentage. It can never be more than 1 or less than 0.

Key idea

In 1654 in France, Blaise Pascal and Pierre de Fermat wrote letters about a puzzle: if a game stops before anyone wins, how should the prize be shared fairly? They solved it by counting the equally likely ways the game could have ended. Their letters helped start probability as a subject.

Words and the Probability Scale

The 1-mark question: pick a word (impossible, unlikely, evens, likely, certain), or mark a cross on a scale from 0 to 1.

A fair spinner has 8 equal parts: 4 stars, 3 moons and 1 sun.

Worked example

1
$P (star) = \frac{4}{8}$ — 4 of the 8 equal parts are stars. The stars go on top and all 8 parts go on the bottom.
2
$\frac{4}{8} = \frac{1}{2}$ — Divide top and bottom by 4. It is exactly half, so the word is evens.
3
$P (star or moon) = \frac{4 + 3}{8} = \frac{7}{8}$ — “Or” means either one counts, so add the 4 stars and 3 moons. 7 eighths is more than a half: likely.
4
$P (sun) = \frac{1}{8}$ — Only 1 part is a sun. That is less than a half but more than 0: unlikely.
5
$P (cloud) = \frac{0}{8} = 0$ — No part shows a cloud, so it cannot happen: impossible.
6
$P (a picture) = \frac{8}{8} = 1$ — All 8 parts show a picture, so it must happen: certain.

Key idea

To mark a cross for $\frac{7}{8}$ on a 0 to 1 scale, split the line into 8 equal parts and count 7 along from 0. It sits just before 1.

Common mistake

Saying a probability is 1.4 or 120%. Nothing can be more likely than certain, and certain is 1. That is the 1-mark "explain" answer.

Count, Then Divide

The 1 or 2-mark question: a bag holds things of different kinds. One is taken at random. Find the probability.

A bag has 7 mint, 5 lemon and 8 cherry sweets.

Worked example

1
$7 + 5 + 8 = 20$ — Add every sweet in the bag. The total goes on the bottom of every answer here.
2
$P (mint) = \frac{7}{20}$ — 7 of the sweets are mint, so 7 goes on top. Each sweet is equally likely to be taken.
3
$7 + 5 = 12$ — “Mint or lemon”: either kind counts, so add the two counts.
4
$P (mint or lemon) = \frac{12}{20}$ — Put the 12 sweets that count over the same total.
5
$7 + 8 = 15$ — “Not lemon” means any of the others, so add the mints and the cherries.
6
$P (not lemon) = \frac{15}{20}$ — 15 sweets count out of 20. Check: $1 - \frac{5}{20}$ is also $\frac{15}{20}$ .

Common mistake

Writing $\frac{7}{13}$ : mint over the other sweets. The bottom is always the total, including the mints.

Key idea

You do not have to simplify. $\frac{12}{20}$ , $\frac{3}{5}$ and 0.6 all get the mark. Never write it as a ratio like 12 : 20.

Probabilities Add Up to 1

The 2 or 3-mark question: a table gives all the probabilities but one. Find the missing one.

Flavour	Probability
banana	0.25
apple	0.4
grape	?
kiwi	0.15

Worked example

1
$0.25 + 0.4 + 0.15 = 0.8$ — Add the three you know: banana, apple and kiwi.
2
$P (all four) = 1$ — One flavour must be picked, which is certain. So the four probabilities add up to 1.
3
$P (grape) = 1 - 0.8 = 0.2$ — Grape is whatever is left of the 1 after the other three.
4
$P (not apple) = 1 - 0.4 = 0.6$ — “Not apple” is every other flavour, so take apple's share away from 1.

Formula

P (not A) = 1 - P (A)

$A$ is any event. $P (A)$ is the chance it happens, and $P (not A)$ is the chance it does not. Together they make 1.

If two missing probabilities are equal, find what is left and halve it: $1 - 0.3 - 0.24 = 0.46$ , so each is 0.23.

Common mistake

Taking the missing numbers from 100 when the table uses decimals. Use 1, not 100.

Either One: Add the Probabilities

The 1 or 2-mark question: find the chance of one thing or another, like red or yellow. If the two cannot happen together, add.

Formula

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

$A$ and $B$ are two outcomes that cannot happen at the same time, like one spin landing on red and on yellow. This is called mutually exclusive.

Colour	red	yellow	purple	green
Probability	0.3	0.15	?	0.25

Worked example

1
$0.3 + 0.15 + 0.25 = 0.7$ — Add the three probabilities you know.
2
$P (purple) = 1 - 0.7 = 0.3$ — The four colours add up to 1, so purple is what is left.
3
$P (red or purple) = 0.3 + 0.3$ — One spin cannot land on both, so add the two probabilities.
4
$= 0.6$ — 0.3 + 0.3 = 0.6. Check: it is 1 take away yellow and green, 1 − 0.4 = 0.6.

Common mistake

Multiplying: 0.3 × 0.3 = 0.09. Multiplying is for two events one after another, like two spins. “Or” on one spin means add.

Key idea

It is the same as adding counts: in the sweet bag, $\frac{7}{20} + \frac{5}{20} = \frac{12}{20}$ for mint or lemon.

Two Gaps: One Twice the Other

The 3 or 4-mark question: a table has two gaps, and you are told how they compare, like “green is twice as likely as red” or a ratio 2 : 1. Use a letter.

Colour	red	blue	green	yellow
Probability	?	0.18	?	0.37

Green is twice as likely as red.

Worked example

1
$P (red) = x, P (green) = 2 x$ — Let x be the smaller one. Twice as likely means green is 2 lots of x.
2
$0.18 + 0.37 = 0.55$ — Add the two probabilities you know.
3
$x + 2 x + 0.55 = 1$ — All four colours add up to 1.
4
$3 x + 0.55 = 1$ — Collect the x terms: x + 2x = 3x.
5
$3 x = 0.45$ — Take 0.55 from both sides.
6
$x = 0.15$ — Divide both sides by 3. So P(red) = 0.15.
7
$P (green) = 2 \times 0.15 = 0.3$ — Green is twice red. Check: 0.15 + 0.18 + 0.3 + 0.37 = 1.

Common mistake

Halving what is left: 0.45 ÷ 2. That only works when the two gaps are equal. x and 2x make 3 equal shares, so divide by 3.

Key idea

Exams often finish by asking how many. If the bag holds 200 beads, green beads $= 0.3 \times 200 = 60$ .

Working Backwards to a Number

The 3 to 5-mark question: you know a probability. Find how many things are in the bag.

A bag has 6 lemon, 9 mint and some cherry sweets. $P (cherry) = \frac{3}{8}$ . How many cherry?

Worked example

1
$6 + 9 = 15$ — Add the sweets you can count. These are the sweets that are not cherry.
2
$1 - \frac{3}{8} = \frac{5}{8}$ — Probabilities add to 1, so the sweets that are not cherry make up 5 eighths of the bag.
3
$\frac{5}{8} of the total = 15$ — Both describe the same sweets: 5 eighths of the bag is 15 sweets.
4
$\frac{1}{8} of the total = 15 \div 5 = 3$ — Divide by 5 to find what one eighth is worth.
5
$total = 8 \times 3 = 24$ — The whole bag is 8 eighths, so multiply one eighth by 8. This is the whole bag.
6
$cherry = 24 - 15 = 9$ — Take the sweets that are not cherry away from the whole bag.
7
$\frac{9}{24} = \frac{3}{8}$ — Check: divide top and bottom by 3. It matches the probability in the question.

Common mistake

Saying there are 3 cherry sweets. $\frac{3}{8}$ is a share of the bag, not a count.

Key idea

Show every line. Most of the marks are for the working (the 15, the $\frac{5}{8}$ , the total), so a slip at the end still scores well. An answer with no working risks scoring nothing.

Adding More to the Bag

The 2 to 4-mark question: more of one kind is put in, and the probability changes. Find how many were added. Hold on to the kind that does not change.

A box has 6 red pens and 10 blue pens. Some more red pens are added. Now $P (red) = \frac{3}{5}$ . How many red pens were added?

Worked example

1
$P (blue) = 1 - \frac{3}{5} = \frac{2}{5}$ — Only red pens were added, so the 10 blue pens stay the same. Blue is the rest of the box.
2
$\frac{2}{5} of the new total = 10$ — The 10 blue pens are 2 fifths of the new box.
3
$\frac{1}{5} of the new total = 10 \div 2 = 5$ — Divide by 2 to find one fifth.
4
$new total = 5 \times 5 = 25$ — The whole box is 5 fifths, so 25 pens now.
5
$red now = 25 - 10 = 15$ — Every pen that is not blue is red.
6
$added = 15 - 6 = 9$ — There were 6 red pens at the start. Check: $\frac{15}{25} = \frac{3}{5}$ .

Common mistake

Keeping the old total of 16. Adding pens makes the box bigger, so the bottom of the fraction changes too.

Key idea

Aiming for $\frac{1}{2}$ ? Half means the two groups are equal. 4 lemon and 12 others: add 8 lemon to make 12 and 12.

Key idea

Like algebra? Call the number added n: $\frac{6 + n}{16 + n} = \frac{3}{5}$ , so $5 (6 + n) = 3 (16 + n)$ , then $30 + 5 n = 48 + 3 n$ and n = 9.

Which Method? Read the Question

Look at what the question gives you.

You see	You do
Choose a word, or mark a cross	make a fraction, compare with a half
Counts of each kind	count over the total
A table with one gap	take the others from 1
“A or B” on one go	add their probabilities
Two gaps, one twice the other	call them x and 2x, add to 1
A probability, find how many	work backwards to the total
More are put in the bag	keep the kind that does not change
Estimate how many times	Relative Frequency and Expectation
A table or chart of results	Probability from Tables and Charts
Two events: A and B	Combined Events with Two-Way Tables

Key idea

Check at the end: your answer must be between 0 and 1.

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