Probability

Find probabilities of single and combined events, use complements and work with and without replacement.

Also known as: chance, independent events, without replacement, Venn diagrams.

SAT · Problem-Solving and Data Analysis GCSE Maths · Probability IGCSE Maths · Probability

Key ideas

P(event) = favourable outcomes ÷ total equally likely outcomes.
P(not A) = 1 − P(A).
For independent events, P(A and B) = P(A) × P(B).
Without replacement, the second probability changes because the total has changed.

Single events

For equally likely outcomes, $P(A) = \frac{\text{favourable}}{\text{total}}$ . On a fair six-sided die, $P(\text{greater than } 4) = \frac{2}{6} = \frac{1}{3}$ .

Probabilities are between $0$ (impossible) and $1$ (certain), and the probabilities of all outcomes sum to $1$ , so $P(\text{not } A) = 1 - P(A)$ .

Combined events

For independent events, multiply. Two fair coins: $P(\text{two tails}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ , so $P(\text{at least one head}) = 1 - \frac{1}{4} = \frac{3}{4}$ .

"At least one" questions are usually easiest with the complement.

Without replacement and two-way data

From a bag of $4$ red and $6$ blue counters, picking two without replacement: $P(\text{both red}) = \frac{4}{10} \times \frac{3}{9} = \frac{2}{15}.$ With overlapping groups (a Venn diagram), avoid double counting: $|A \cup B| = |A| + |B| - |A \cap B|$ .

Worked examples

Example 1 · Without replacement

A bag contains 4 red and 6 blue counters. Two are taken at random without replacement. Find P(both red).

Step 1.First red: $\frac{4}{10}$ .
Step 2.Second red (one red removed): $\frac{3}{9}$ .
Step 3.Multiply: $\frac{12}{90} = \frac{2}{15}$ .

Answer: $\frac{2}{15}$

Example 2 · Overlapping groups

Of 200 students, 120 study Spanish, 70 study French and 30 study both. One student is chosen at random. Find P(neither).

Step 1.At least one language: $120 + 70 - 30 = 160$ .
Step 2.Neither: $200 - 160 = 40$ .
Step 3. $P = \frac{40}{200} = 0.2$ .

Answer: 0.2

Common mistakes

Adding probabilities that should be multiplied (and vice versa).
Keeping the same denominator after an item is removed.
Counting the overlap twice in Venn diagram problems.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1ProbabilityEasier

A fair six-sided die is rolled. What is the probability of rolling a number greater than 4?

Question 2ProbabilityEasier

A bag contains 3 red, 5 blue and 2 green marbles. One marble is chosen at random. What is the probability that it is blue?