Summary Revision Notes for Grade 11 NSC Matric Mathematics

Summary

Key terminology

Understanding probability requires mastering specific terms that describe different aspects of experiments and their outcomes.

Outcome: This is one single result from an experiment. For example, when you flip a coin, getting "heads" is one outcome.

Sample space: This represents all the different things that could happen in an experiment. When rolling a six-sided dice, the sample space includes $\{1, 2, 3, 4, 5, 6\}$ .

Event: An event is a collection of one or more outcomes from an experiment. For instance, rolling an even number on a dice is an event that includes outcomes $\{2, 4, 6\}$ .

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The distinction between outcomes and events is crucial: an outcome is a single result, while an event can contain multiple outcomes.

Probability of an event: This is a number between 0 and 1 that tells us how likely an event is to happen. A probability of 0 means impossible, while 1 means certain.

Relative frequency: This measures how often an event occurs during repeated trials of an experiment. You calculate it by dividing the number of times the event happened by the total number of trials.

Types of events

Union of events: Written as "A or B", this includes all outcomes that appear in at least one of the events. Think of it as combining the events together.

Intersection of events: Written as "A and B", this includes only the outcomes that appear in both events simultaneously.

Mutually exclusive events: These are events that cannot happen at the same time. They have no outcomes in common, so $(A \text{ and } B) = \emptyset$ . For example, getting both heads and tails on a single coin flip is impossible.

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Critical Concept: Mutually exclusive events have NO overlap - they share zero outcomes in common.

Complementary events: These are two mutually exclusive events that together include every possible outcome in the sample space. We write the complement of event A as "not A".

Independent events: Two events are independent when knowing the result of one event doesn't change the probability of the other event. For independence: $P(A \text{ and } B) = P(A) \times P(B)$ .

Essential probability rules

These mathematical rules help you calculate probabilities in different situations.

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Addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

This rule works for any two events. You add the individual probabilities but subtract the overlap to avoid counting it twice.

Addition rule for mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$

When events cannot happen together, there's no overlap to subtract, so you simply add the probabilities.

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

The probability that an event doesn't happen equals 1 minus the probability that it does happen. This is useful when it's easier to calculate the opposite outcome.

Visual tools for probability

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Venn diagram: This visual tool shows how events overlap using circles or shapes. Each region represents an event and can show the outcomes, number of outcomes, or probability of that event. Venn diagrams help you see relationships between events clearly.

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Tree diagram: This tool displays dependent events as branches of a tree. Each branch shows an outcome with its probability. To find the probability of a specific combination:

Find all paths leading to your desired outcome
Multiply the probabilities along each complete path
Add the probabilities from different paths together

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Two-way contingency table: This organises data when you have two events, each with only two possible outcomes. The table shows counts for every combination of outcomes, plus row and column totals. This helps determine if events are dependent or independent.

Worked examples

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Worked Example: Addition Rule

In a class of 30 students, 18 play football, 12 play rugby, and 8 play both sports. What's the probability a randomly selected student plays at least one sport?

Solution:

$P(\text{football}) = \frac{18}{30} = \frac{3}{5}$
$P(\text{rugby}) = \frac{12}{30} = \frac{2}{5}$
$P(\text{both}) = \frac{8}{30} = \frac{4}{15}$
$P(\text{football or rugby}) = \frac{3}{5} + \frac{2}{5} - \frac{4}{15} = \frac{9}{15} + \frac{6}{15} - \frac{4}{15} = \frac{11}{15}$

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Worked Example: Complementary Rule

The probability of rain tomorrow is 0.3. What's the probability it won't rain?

Solution: $P(\text{no rain}) = 1 - P(\text{rain}) = 1 - 0.3 = 0.7$

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Worked Example: Independent Events

Two dice are rolled. What's the probability of getting a 6 on both dice?

Solution: Since the dice don't influence each other, they're independent.

$P(\text{first die shows 6}) = \frac{1}{6}$
$P(\text{second die shows 6}) = \frac{1}{6}$
$P(\text{both show 6}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

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Key Points to Remember:

Probability values always fall between 0 and 1 inclusive
Mutually exclusive events cannot happen simultaneously, so you just add their probabilities
Independent events don't affect each other, so you multiply their probabilities
Complementary events always add up to 1
Visual tools like Venn diagrams and tree diagrams make complex probability problems easier to solve

Summary (Grade 11 NSC Matric Mathematics): Revision Notes

Summary

Key terminology

Types of events

Essential probability rules

Visual tools for probability

Worked examples

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