IEB Grade 12 Probability and Counting Cheat Sheet

Probability and counting carry about 15 marks of Paper 1. Every rule and method is here.

Rules

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

• Mutually exclusive: $P(A\text{ and }B)=0$. Complement: $P(\text{not }A)=1-P(A)$.

• Independent: $P(A\text{ and }B)=P(A)\times P(B)$. Use it as the test.

Diagrams and tables

• Venn: fill from the middle outwards; the total is 1 (or $n$).

• Tree: multiply along a path, add between paths. Without replacement the second-stage numbers change.

• Contingency table: divide by the grand total; compare $P(A)P(B)$ with $P(A\text{ and }B)$.

Counting

• Counting principle: multiply the options for each position. Fill restricted positions first.

• $n!$ arrangements of $n$ different items; $0!=1$.

• Repeated items: $\frac{n!}{a!b!\dots }$

• Kept together: treat as a block, then arrange inside the block.

• Kept apart: total $-$ together.

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

Know by heart

• Independent versus mutually exclusive

• The block and complement methods

On the formula sheet

• $P(A)=\frac{n(A)}{n(S)}$

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

Common traps

• Treating mutually exclusive events as independent.

• Forgetting to arrange items inside a block.

• Not dividing by the factorial of repeated letters.

• Conditional probability notation: it is not in the syllabus.

Related topics

• probability rules
• tree diagrams and contingency tables
• fundamental counting principle
• arrangements with restrictions