Summary (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

Key probability rules

Addition rule

The addition rule (also called the sum rule) helps us find the probability of either event A or event B occurring. This fundamental rule connects individual event probabilities with their combined probability.

Addition rule for mutually exclusive events

When two events are mutually exclusive (they cannot happen at the same time), the addition rule becomes much simpler. This is because mutually exclusive events have no overlap, so $P(A \text{ and } B) = 0$.

Complementary rule

The complementary rule is one of the most useful probability rules, especially for solving "at least one" problems. It states that the probability of an event not occurring is:

$P(\text{not } A) = 1 - P(A)$

This rule works because A and (not A) are complementary events - they cover all possible outcomes and cannot happen together, so their probabilities add up to 1.

Product rule for independent events

When two events A and B are independent (one does not affect the other), calculating the probability of both occurring is straightforward:

$P(A \text{ and } B) = P(A) \times P(B)$

Product rule for dependent events

When two events A and B are dependent (one affects the probability of the other), the simple multiplication rule doesn't work:

$P(A \text{ and } B) \neq P(A) \times P(B)$

Visual tools for probability

Visual representation is crucial for understanding complex probability problems. These tools help organise information and prevent calculation errors.

Venn diagrams

Venn diagrams show how events relate to each other using shapes (usually circles or rectangles). Each event is represented by a shape, with the region inside representing outcomes included in that event. The region outside represents outcomes not in the event. Venn diagrams are particularly helpful for visualising unions, intersections, and complements of events.

Tree diagrams

Tree diagrams help organise and visualise different possible outcomes in a sequence of events. Each branch shows an outcome with its probability written alongside. For each possible outcome of the first event, we draw branches for all possible outcomes of the second event. The probability of any complete sequence is calculated by multiplying the probabilities along the branches.

Two-way contingency tables

Two-way contingency tables record counts or percentages for probability problems involving two variables. These tables are especially useful for determining whether events are dependent or independent by comparing observed frequencies with expected frequencies.

Counting principles

Counting principles form the foundation of probability calculations when dealing with equally likely outcomes. Understanding these principles is essential for solving complex probability problems.

Fundamental counting principle

The fundamental counting principle states that if there are $n(A)$ outcomes for event A and $n(B)$ outcomes for event B, then there are $n(A) \times n(B)$ different possible outcomes for both events combined.

Multiple choices with replacement

When you have n objects to choose from and you choose r times, with the number of choices remaining the same after each choice (replacement), the total number of possibilities is:

$n \times n \times n \times \ldots \times n \text{ (r times)} = n^r$

Arrangements without repetition

The number of arrangements of n different objects is:

$n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1 = n!$

This is called "n factorial" and represents the number of ways to arrange n distinct objects in order.

Arrangements with repetition

For a set of n objects where there are k subsets with repeated objects ($n_1$ are the same, $n_2$ are the same, ..., $n_k$ are the same), the number of arrangements is:

$\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}$

This formula accounts for the fact that identical objects cannot be distinguished when arranged.