Revision (Grade 12 NSC Matric Mathematics): Revision Notes

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Key terminology

Understanding probability begins with knowing the fundamental terms that form the foundation of all probability problems.

Outcome

An outcome represents a single result from an uncertain or random process (called an experiment). When you accidentally drop a book, the possible outcomes include it falling on its cover, back, or side. Each of these represents one possible outcome from the dropping experiment.

Sample space

The sample space is the complete collection of all possible outcomes for any given experiment. We use the letter $S$ to represent this set. For instance, when rolling a standard six-sided die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$. Every experiment has exactly one sample space containing all its possible results.

Event

An event consists of a collection of outcomes from an experiment. Using a standard 52-card deck as an example, one event might be "drawing a spade card" whilst another could be "drawing a king card". Events are essentially subsets of the sample space.

Probability of an event

The probability of an event is a numerical value between 0 and 1 (inclusive) that measures how likely the event is to occur. A probability of 0 indicates the event will never happen, whilst a probability of 1 means it will always happen.

To calculate probability, divide the number of favourable outcomes by the total number of possible outcomes in the sample space. This gives us the basic probability formula:

$P(\text{Event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$

Relative frequency

Relative frequency refers to how often an event actually occurs during repeated experimental trials, expressed as a fraction of the total trials conducted. For example, if you flip a coin 10 times and get heads 3 times, the relative frequency of heads is $\frac{3}{10} = 0.3$.

Union of events

The union of events represents all outcomes that occur in at least one of the specified events. For two events $A$ and $B$, we write this as "$A$ or $B$" or use set notation $A \cup B$. If $A$ represents African countries and $B$ represents European countries, then $A$ or $B$ includes all countries from either Africa or Europe (or both).

Intersection of events

The intersection of events includes only those outcomes that occur in all specified events simultaneously. For events $A$ and $B$, we write this as "$A$ and $B$" or use set notation $A \cap B$. If $A$ represents soccer players and $B$ represents cricket players, then $A$ and $B$ refers to people who play both sports.

Mutually exclusive events

Mutually exclusive events are events that share no common outcomes - they cannot happen at the same time. These events form an empty set when intersected $(A \text{ and } B) = \emptyset$. A classic example is the event that a number is even versus the event that the same number is odd - these are mutually exclusive since no number can be both even and odd simultaneously.

Complementary events

Complementary events are two mutually exclusive events that together include every possible outcome in the sample space. For any event $A$, we write its complement as "not $A$" or $A'$. These events are special because they cover all possibilities without overlapping.

Independent and dependent events

Two events $A$ and $B$ are independent when the outcome of the first event has no influence on the outcome of the second event. Coin flips provide a perfect example - if you flip a coin and get tails, this doesn't affect whether your next flip will be heads or tails.

Conversely, events are dependent when one event's outcome influences the other's outcome. Consider a lunchbox containing 3 sandwiches and 2 apples - when you eat one item, this reduces your remaining choices and affects the probability of what you might eat next.

Essential probability rules

These rules form the mathematical foundation for calculating probabilities in various scenarios.

Addition rule

The addition rule (also called the sum rule) helps us find the probability of either event $A$ or event $B$ occurring:

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

This formula accounts for the probabilities of individual events whilst subtracting the overlap (intersection) to avoid double-counting.

Addition rule for mutually exclusive events

When two events are mutually exclusive, they cannot occur simultaneously, so $P(A \text{ and } B) = 0$. The addition rule simplifies to:

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

Complementary rule

The complementary rule states that an event and its complement always have probabilities that sum to 1:

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

This rule proves particularly useful when calculating the probability of "at least one" scenarios.

Product rule for independent events

When two events are independent, the probability of both occurring equals the product of their individual probabilities:

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

Product rule for dependent events

When events are dependent, the product rule doesn't apply in its simple form:

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

Instead, we need conditional probability concepts to handle dependent events properly.