Practice Maths

Probability Rules

Key Terms

The addition rule gives the probability of A or B: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
For mutually exclusive events (A ∩ B = ∅): P(A ∪ B) = P(A) + P(B). There is no overlap to subtract.
Complementary events
: P(A′) = 1 − P(A). Together, A and A′ cover the entire sample space.
A two-way table (also called a contingency table) organises data by two categorical variables. Each cell count divided by the total gives a probability.
Marginal probabilities
are found from row or column totals. Joint probabilities come from individual cells.
Always check: all probabilities in a valid table must be between 0 and 1, and sum to 1.
Probability Rules — Summary
• Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Mutually exclusive: P(A ∪ B) = P(A) + P(B)  [when A ∩ B = ∅]
• Complement: P(A′) = 1 − P(A)
• Partition rule: P(A) = P(A ∩ B) + P(A ∩ B′)
• From a two-way table: P(row ∩ column) = cell frequency ÷ grand total

Example two-way table: 100 students surveyed about preferred sport and gender.

Football Swimming Total
Male 32 18 50
Female 20 30 50
Total 52 48 100

P(Male ∩ Football) = 32/100. P(Female) = 50/100. P(Swimming) = 48/100.

Hot Tip When using the addition rule, the most common mistake is forgetting to subtract the intersection. Think: if you add P(A) and P(B), the overlap is counted twice, so you subtract it once. If the events are mutually exclusive (no overlap), there is nothing to subtract. Also: always check P(A′) = 1−P(A) as a quick sanity check.

Worked Example 1 — Applying the addition rule

Question: P(A) = 0.45, P(B) = 0.38, P(A ∩ B) = 0.15. Find P(A ∪ B) and P(A′ ∩ B′).

P(A ∪ B) = 0.45 + 0.38 − 0.15 = 0.68

P(A′ ∩ B′) = 1 − P(A ∪ B) = 1 − 0.68 = 0.32

Note: De Morgan’s Law: (A ∪ B)′ = A′ ∩ B′, so “neither A nor B” = 1 − P(A ∪ B).

Worked Example 2 — Two-way table

Question: Using the table above, find: (i) P(Female ∪ Swimming), (ii) P(Male ∩ Swimming′).

(i) P(Female ∪ Swimming) = P(Female) + P(Swimming) − P(Female ∩ Swimming)

= 50/100 + 48/100 − 30/100 = 68/100 = 0.68

(ii) P(Male ∩ Swimming′) = P(Male ∩ Football) = 32/100 = 0.32

What Is Probability?

Probability is a number between 0 and 1 that measures how likely an event is to occur. A probability of 0 means impossible; a probability of 1 means certain. We often express probability as a fraction, decimal, or percentage.

The sample space S is the set of all possible outcomes. An event is any subset of the sample space. For a fair experiment, the probability of an event is:

P(A) = (number of outcomes in A) ÷ (total number of outcomes in S)

The Addition Rule — Probability of A or B

When we want P(A or B), we need to be careful about double-counting. If A and B share outcomes (they overlap), those shared outcomes get counted in both P(A) and P(B). We must subtract them once:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive — meaning they cannot both happen at the same time (no overlap) — then P(A ∩ B) = 0 and the rule simplifies to:

P(A ∪ B) = P(A) + P(B)  [mutually exclusive only]

Do not apply the simplified rule carelessly: Only omit the P(A ∩ B) term when the events genuinely cannot co-occur. If a question does not explicitly say the events are mutually exclusive, check whether they overlap before simplifying.

Complementary Events

The complement of event A, written A′ (or A̅ or ¬A), is the event that A does not happen. Since every outcome is either in A or not in A:

P(A′) = 1 − P(A)

This is one of the most useful shortcuts in probability. Whenever calculating P(at least one) or P(not A), it is often easier to calculate P(A) and subtract from 1.

Example: The probability of rolling at least one six in two rolls of a die. P(no six on one roll) = 5/6. P(no six in two rolls) = (5/6)2 = 25/36. So P(at least one six) = 1 − 25/36 = 11/36.

Two-Way Tables

A two-way table (contingency table) organises data by two categorical variables. It is an extremely efficient tool for finding probabilities because all the information is laid out in one place.

Reading a two-way table:

  • Cell values give the frequency (count) for a specific combination of categories.
  • Row totals and column totals (marginals) give the frequency for one category alone.
  • To find a probability, divide any frequency by the grand total (bottom-right cell).
  • Joint probabilities (e.g. P(Male ∩ Football)) come from individual cells ÷ grand total.
  • Marginal probabilities (e.g. P(Male)) come from row/column totals ÷ grand total.

The Partition Rule

Any event A can be split into two non-overlapping parts using another event B:

P(A) = P(A ∩ B) + P(A ∩ B′)

This is useful when you can only calculate partial intersection probabilities but need the total probability of A.

Exam tip: Always draw a Venn diagram or two-way table if the problem involves more than one event. It takes 30 seconds and eliminates most errors — particularly the mistake of forgetting to subtract P(A ∩ B) in the addition rule.

Mastery Practice

See Answers ➔
  1. Fluency

    Apply the addition rule. Find P(A ∪ B) in each case.

    1. (a) P(A) = 0.3, P(B) = 0.5, P(A ∩ B) = 0.1
    2. (b) P(A) = 1/4, P(B) = 1/3, P(A ∩ B) = 1/12
    3. (c) P(A) = 0.6, P(B) = 0.7, P(A ∩ B) = 0.4
    4. (d) A and B are mutually exclusive: P(A) = 0.25, P(B) = 0.35
  2. Fluency

    Use the complement rule. Find the missing probability.

    1. (a) P(A) = 0.72. Find P(A′).
    2. (b) P(A′) = 3/7. Find P(A).
    3. (c) The probability it does NOT rain tomorrow is 0.35. Find the probability it does rain.
    4. (d) A student passes a test with probability 4/5. Find the probability the student fails.
  3. Fluency

    Use this two-way table of 80 students (rows: Year 11, Year 12; columns: Science, Arts).

    Year 11/Science: 18  |  Year 11/Arts: 22  |  Year 12/Science: 24  |  Year 12/Arts: 16  |  Total: 80

    1. (a) Find P(Year 11).
    2. (b) Find P(Science).
    3. (c) Find P(Year 12 ∩ Arts).
    4. (d) Find P(Year 11 ∪ Science).
  4. Fluency

    For each situation, identify whether the events are mutually exclusive and find P(A ∪ B).

    1. (a) Drawing a club or a diamond from a standard deck.
    2. (b) Drawing a heart or a queen from a standard deck.
    3. (c) Rolling a prime number or an even number on a die.
  5. Understanding

    Two-way table — complete the missing values.

    Method: Row and column totals must be consistent. Use arithmetic to find missing cells.

    100 people surveyed (Employed/Unemployed × Male/Female). Male/Employed = 35. Male total = 55. Female/Unemployed = 18. Grand total = 100.

    1. (a) Complete the full two-way table (find all four cells and both marginal totals).
    2. (b) Find P(Employed).
    3. (c) Find P(Female ∩ Employed).
    4. (d) Find P(Male ∪ Employed).
  6. Understanding

    Finding an unknown probability using the addition rule.

    1. (a) P(A ∪ B) = 0.75, P(A) = 0.50, P(B) = 0.40. Find P(A ∩ B).
    2. (b) P(A) = 0.6, P(A ∩ B) = 0.24. If A and B are not mutually exclusive and P(A ∪ B) = 0.82, find P(B).
    3. (c) P(A ∪ B) = 0.9 and A and B are mutually exclusive with P(A) = 3P(B). Find P(A) and P(B).
  7. Understanding

    Multi-step probability problem.

    In a group of 50 students: P(studies Maths) = 0.6, P(studies English) = 0.7, and 10 students study neither subject.

    1. (a) Find P(Maths ∪ English).
    2. (b) Find P(Maths ∩ English).
    3. (c) Find the number of students who study both subjects.
    4. (d) Find the number who study Maths only.
  8. Understanding

    Partition rule and total probability.

    In a factory, 60% of products come from Machine A and 40% from Machine B. 5% of Machine A’s products are defective and 8% of Machine B’s products are defective.

    1. (a) Using the partition rule, find P(Defective).
    2. (b) Find P(Not defective).
    3. (c) Find P(Machine A ∩ Defective).
    4. (d) Are “From Machine A” and “Defective” mutually exclusive? Explain.
  9. Problem Solving

    Constructing a two-way table from conditional information.

    Challenge. 200 people were surveyed. 120 are under 30 (U30) and 80 are 30 or over (30+). Among the U30 group, 75 own a smartphone. Among the 30+ group, 48 own a smartphone.
    1. (a) Construct a complete two-way table (U30/30+ × Smartphone/No smartphone).
    2. (b) Find P(Smartphone).
    3. (c) Find P(U30 ∪ Smartphone).
    4. (d) Find P(30+ ∩ No smartphone).
  10. Problem Solving

    Probability rules — multi-event reasoning.

    Challenge.
    1. (a) Three mutually exclusive events A, B, C satisfy P(A) = 2k, P(B) = 3k, P(C) = 4k and together they cover the entire sample space. Find k and all three probabilities.
    2. (b) P(A) = 0.5, P(B) = 0.4. If P(A ∪ B) = 0.7, find P(A ∩ B). Then use De Morgan’s Law to find P(A′ ∪ B′).
    3. (c) Show algebraically that P(A only) + P(B only) + P(A ∩ B) = P(A ∪ B). Start from the definition that A only = A ∩ B′.