Probability laws

Objective. Use complements, unions and intersections consistently.

1. Learn
2. Worked example
3. Practice
4. Feedback
5. Continue

Learn: the key idea

Probability laws give consistent rules for combining event probabilities, where the complement rule states P(A') = 1 − P(A), the addition rule for any two events states P(A∪B) = P(A) + P(B) − P(A∩B), and for mutually exclusive events P(A∩B) = 0 so the formula simplifies to P(A∪B) = P(A) + P(B). The subtraction of P(A∩B) in the general addition rule avoids double-counting outcomes that satisfy both events. For example, if P(A) = 0.4, P(B) = 0.5 and P(A∩B) = 0.2, then P(A∪B) = 0.4 + 0.5 − 0.2 = 0.7. These laws underpin almost all further probability calculations including conditional probability and independence.

Worked example

Given P(A) = 0.3, P(B) = 0.6 and P(A∩B) = 0.15, use the probability laws to find P(A∪B).

1 $Recall the addition law: P(A\cupB) = P(A) + P(B) - P(A\capB).$
2 $Substitute the values: P(A\cupB) = 0.3 + 0.6 - 0.15.$
3 $Add the first two: 0.3 + 0.6 = 0.9.$
4 $Subtract the intersection: 0.9 - 0.15 = 0.75.$

Answer: 0.75

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students frequently add P(A) + P(B) without subtracting P(A∩B) even when the events overlap, double-counting outcomes in both events. Always check whether events are mutually exclusive before omitting the intersection term.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard