Probability: Core Rules — notes, question types & practice · Permutation, Combination & Probability

01

What probability means

Probability measures how likely an event is.

P(E) = \frac{\text{favourable outcomes}}{\text{total outcomes}}

The value always lies between 0 and 1. Zero means impossible; one means certain.

A bag has 4 red and 6 blue balls. One draw: $P(\text{red}) = \dfrac{4}{10} = \dfrac{2}{5}$ .

02

The sample space

The sample space is the list of all equally likely outcomes. Count it first.

One die: 6 outcomes. Two dice: $6 \times 6 = 36$ . One card: 52. Three coins: $2^3 = 8$ .

Rule: Every probability in a question must sit on the same sample space.

03

The complement rule

"NOT the event" is its complement.

P(\text{not } E) = 1 - P(E)

A number from 1 to 20: primes number 8, so $P(\text{prime}) = \dfrac{8}{20}$ and $P(\text{not prime}) = \dfrac{12}{20} = \dfrac{3}{5}$ .

04

The addition rule

For "A or B", add the two probabilities and remove the overlap counted twice.

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

A card: $P(\text{king}) = \dfrac{4}{52}$ , $P(\text{heart}) = \dfrac{13}{52}$ , and the king of hearts sits in both groups, so $P = \dfrac{4 + 13 - 1}{52} = \dfrac{16}{52} = \dfrac{4}{13}$ .

If A and B cannot happen together, the overlap is 0 and the last term drops.

05

The multiplication rule

For "A and B" in sequence, multiply.

Two shooters hit with $\dfrac{1}{2}$ and $\dfrac{1}{3}$ , independently. Both hit: $\dfrac{1}{2} \times \dfrac{1}{3} = \dfrac{1}{6}$ .

Without replacement, the second draw's total shrinks. A bag with 3 red and 2 white, two draws, both red: $\dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{3}{10}$ .

Rule: With replacement, totals stay. Without replacement, each total drops by one.

06

At least one

"At least one" means one or more. Counting each case is slow. Flip it:

P(\text{at least one}) = 1 - P(\text{none})

Two shots with $\dfrac{1}{2}$ and $\dfrac{1}{3}$ : none hit $= \dfrac{1}{2} \times \dfrac{2}{3} = \dfrac{1}{3}$ , so at least one hit $= \dfrac{2}{3}$ .

Tip: The complement of "at least one" is exactly "none". Use this pair every time.

07

Drawing without replacement

Two draws together behave like one combined selection. Two roads, one answer.

A bag holds 4 red and 3 black. Both black: multiply $\dfrac{3}{7} \times \dfrac{2}{6} = \dfrac{1}{7}$ , or select $^{3}C_2 \div \, ^{7}C_2 = \dfrac{3}{21} = \dfrac{1}{7}$ .

One of each colour: $\dfrac{^{4}C_1 \times ^{3}C_1}{^{7}C_2} = \dfrac{12}{21} = \dfrac{4}{7}$ .

Example: Order makes no difference in a simultaneous draw — count unordered pairs with nCr and both roads agree.

08

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Single-event probability

How to spot it:

One draw or one pick: 'a ball is drawn', 'a number is chosen from 1 to 30'.

P(E) = \frac{\text{favourable}}{\text{total}}

Method

Count the total outcomes.
Count the favourable outcomes.
Write the fraction and reduce it.

Why it works:

Equally likely outcomes make probability one favourable share of the whole.

Try this

A bag has 5 red and 3 white balls. One ball is drawn. Find P(red).

Show solution

Total 8, red 5.
$\dfrac{5}{8}$ .

Answer

5/8

Type 2very common2 practice Q

Complement (not the event)

How to spot it:

'…is not a prime', 'does not happen', or the direct count looks long.

P(\text{not } E) = 1 - P(E)

Method

Find $P(E)$ the short way.
Subtract from 1.
Reduce the fraction.

Why it works:

The event and its complement split all outcomes, so their probabilities add to 1.

Try this

A number is chosen from 1 to 20. Find the probability that it is not prime.

Show solution

Primes: 8, so $P(\text{prime}) = \dfrac{8}{20}$ .
$1 - \dfrac{2}{5} = \dfrac{3}{5}$ .

Answer

3/5

Type 3common2 practice Q

Either-or (addition rule)

How to spot it:

'King or heart', 'red or blue', 'A or B' — the two groups may overlap.

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Method

Add the two probabilities.
Subtract the overlap once.
No overlap? Skip the subtraction.

Why it works:

Outcomes in both events get counted twice, so the overlap is removed once.

Try this

One card is drawn from a deck. Find P(king or heart).

Show solution

$\dfrac{4}{52} + \dfrac{13}{52} - \dfrac{1}{52}$ .
$= \dfrac{16}{52} = \dfrac{4}{13}$ .

Answer

4/13

Type 4common2 practice Q

Independent events (multiplication)

How to spot it:

'With replacement', 'independently', or two separate shooters, coins or exams.

P(A \cap B) = P(A) \times P(B)

Method

Confirm the events do not affect each other.
Multiply the probabilities.
Reduce the fraction.

Why it works:

Independence leaves each probability unchanged, so the joint chance is the product.

Try this

Two shooters hit a target with probabilities 1/2 and 1/3, independently. Find P(both hit).

Show solution

$\dfrac{1}{2} \times \dfrac{1}{3}$ .
$= \dfrac{1}{6}$ .

Answer

1/6

Type 5very common2 practice Q

At least one via the complement

How to spot it:

'At least one hit / head / six', 'at least one red ball is drawn'.

1 - P(\text{none})

Method

Find $P(\text{none})$ — the complement case.
Subtract from 1.
Reduce the fraction.

Why it works:

'None' is a single clean case, while 'at least one' splits into many.

Try this

Two shooters hit with probabilities 3/4 and 2/3, independently. Find P(at least one hits).

Show solution

None: $\dfrac{1}{4} \times \dfrac{1}{3} = \dfrac{1}{12}$ .
$1 - \dfrac{1}{12} = \dfrac{11}{12}$ .

Answer

11/12

Type 6very common3 practice Q

Draws without replacement

How to spot it:

'Two balls are drawn together / one after another' and nothing is put back.

Method

Multiply draw by draw, shrinking the totals.
Or count pairs with nCr and divide.
For 'one of each colour', pick one from each colour and multiply.

Why it works:

Every draw removes a ball, so each later draw runs on a smaller pool.

Try this

A bag has 5 red and 4 white balls. Two are drawn without replacement. Find P(both red).

Show solution

$\dfrac{5}{9} \times \dfrac{4}{8}$ .
$= \dfrac{5}{18}$ .

Answer

5/18

09

Formula sheet

Classical definition

P(E) = \frac{m}{n}

$m$ favourable of $n$ equally likely outcomes.

Complement

P(\bar{E}) = 1 - P(E)

'Not the event'.

Addition rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Subtract the overlap once.

Multiplication (independent)

P(A \cap B) = P(A) \times P(B)

For independent events.

At least one

P(\text{at least one}) = 1 - P(\text{none})

The complement of 'none'.

10

Shortcuts that save time

⚡ Flip 'at least' to 'none'

For any 'at least one' question, compute the none case and subtract from 1. It is always shorter.

Example

Two dice are thrown. Find P(at least one six).

Show solution

None: $\dfrac{25}{36}$ .
$1 - \dfrac{25}{36} = \dfrac{11}{36}$ .

Answer

11/36

⚡ Multiply draw by draw

Without replacement, multiply the fractions and let each total drop by one. With replacement, totals stay.

Example

A bag has 4 red and 3 black balls. Two are drawn without replacement. Find P(both black).

Show solution

$\dfrac{3}{7} \times \dfrac{2}{6}$ .
$= \dfrac{1}{7}$ .

Answer

1/7

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Adding probabilities of overlapping events — 'king or heart' must remove the king of hearts once.

Mistake 02

Multiplying without shrinking the total — without replacement the second draw runs on one ball fewer.

Mistake 03

Counting 'at least one' case by case — use $1 - P(\text{none})$ instead.

Mistake 04

Reporting a probability above 1 — a probability always lies between 0 and 1, so recheck the sample space.

Mistake 05

Mixing 'exactly one' with 'at least one' — they are different events with different counts.

12

Quick revision

Read this the night before the exam.

$P = \dfrac{\text{favourable}}{\text{total}}$ , always between 0 and 1.
Complement: $P(\text{not } E) = 1 - P(E)$ .
Addition: $P(A) + P(B) - P(A \cap B)$ .
Independent 'and': multiply the probabilities.
Without replacement: each total drops by one.
At least one $= 1 - P(\text{none})$ .

13

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.