Definition of Probability Revision Notes for HSC SSCE Mathematics Standard

Definition of Probability

Introduction to probability

Probability measures how likely it is that a particular event will occur. To calculate probability accurately, we use a formal mathematical definition. When you perform a random experiment, the outcome or result is called an event.

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For example, tossing a coin is a random experiment, and getting a head is an event. Understanding the difference between the experiment (the action) and the event (the outcome) is fundamental to probability theory.

Probability can be expressed in three different ways: as a fraction, as a decimal, or as a percentage. Understanding how to work with all three forms is important for solving probability problems.

Understanding events and experiments

A random experiment is an activity where the outcome cannot be predicted with certainty. Examples include tossing a coin, rolling a die, or selecting a card from a deck.

An event is a specific outcome or result from a random experiment. For instance, if you toss a coin, getting heads is one event and getting tails is another event.

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Probability Notation:

We use special notation to represent probability:

The letter $E$ denotes an event
$P(E)$ refers to the probability that event E occurs
$n(E)$ represents the number of favourable outcomes
$n(S)$ represents the total number of possible outcomes (the sample space)

Remember: the letter $n$ stands for "number of" in this notation.

The probability formula

The probability of an event is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. This can be written mathematically as:

$P(E) = \frac{n(E)}{n(S)}$

In words, this means:

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

The favourable outcomes are the outcomes you are interested in, whilst the total outcomes include all possible results from the experiment.

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Think of it as: "Favourable over Total"

This simple memory aid helps you remember which number goes on top (favourable outcomes) and which goes on the bottom (total outcomes) when calculating probability.

Worked example: Calculating probability with coins

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Worked Example: Coin Selection Probability

Imagine you have a collection of 10 coins: 7 one dollar coins (ODC) and 3 two dollar coins (TDC). You select one coin at random. What is the probability that the coin is:

Part a: A one dollar coin

Start by writing the probability formula: $P(\text{ODC}) = \frac{n(\text{ODC})}{n(S)}$
Identify the favourable outcomes: there are 7 one dollar coins
Identify the total outcomes: there are 10 coins in total
Substitute these values into the formula: $P(\text{ODC}) = \frac{7}{10}$
The fraction is already in simplest form
Convert to decimal form: $P(\text{ODC}) = 0.7$
Convert to percentage form: $P(\text{ODC}) = 70\%$

Part b: A two dollar coin

Write the probability formula: $P(\text{TDC}) = \frac{n(\text{TDC})}{n(S)}$
Identify the favourable outcomes: there are 3 two dollar coins
Identify the total outcomes: there are 10 coins in total
Substitute into the formula: $P(\text{TDC}) = \frac{3}{10}$
The fraction is already in simplest form
Convert to decimal form: $P(\text{TDC}) = 0.3$
Convert to percentage form: $P(\text{TDC}) = 30\%$

Notice: The two probabilities add up to 1 (or 100%). This makes sense because you must choose either a one dollar coin or a two dollar coin.

Worked example: Calculating probability with playing cards

Before working with playing cards, you need to understand their structure. A standard deck of playing cards contains 52 cards divided into four suits: clubs, spades, hearts, and diamonds. Each suit contains 13 cards ranging from ace to king. Within each suit, there are three picture cards: the jack, queen, and king. This means there are 12 picture cards in total (3 picture cards × 4 suits).

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Worked Example: Playing Card Probabilities

Let's calculate the probability of selecting specific cards from a standard deck.

Part a: Red four

Write the formula: $P(\text{Red 4}) = \frac{n(\text{Red 4})}{n(S)}$
Count the favourable outcomes: there are 2 red fours in the deck (one in hearts and one in diamonds)
Count the total outcomes: there are 52 cards in total
Substitute: $P(\text{Red 4}) = \frac{2}{52}$
Simplify the fraction by dividing both numerator and denominator by 2: $P(\text{Red 4}) = \frac{1}{26}$

Part b: Diamond

Write the formula: $P(\text{Diamond}) = \frac{n(\text{Diamond})}{n(S)}$
Count the favourable outcomes: there are 13 diamonds in the deck
Count the total outcomes: there are 52 cards in total
Substitute: $P(\text{Diamond}) = \frac{13}{52}$
Simplify by dividing both numerator and denominator by 13: $P(\text{Diamond}) = \frac{1}{4}$

Part c: Picture card

Write the formula: $P(\text{Picture}) = \frac{n(\text{Picture})}{n(S)}$
Count the favourable outcomes: there are 12 picture cards in the deck
Count the total outcomes: there are 52 cards in total
Substitute: $P(\text{Picture}) = \frac{12}{52}$
Simplify by dividing both numerator and denominator by 4: $P(\text{Picture}) = \frac{3}{13}$

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Exam tip: When working with playing cards, always check your work by considering whether your answer makes sense. For example, the probability of selecting a diamond should be $\frac{1}{4}$ because diamonds make up one quarter of the deck.

Equally likely outcomes

Equally likely outcomes occur when there is no reason to expect one outcome to happen more often than any other. Each outcome has the same chance of occurring.

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Example of equally likely outcomes:

If you select a ball at random from a bag containing one red ball, one blue ball, and one white ball, each ball has an equal probability of being chosen. There is no reason why one colour would be selected more often than another.

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Example of outcomes that are NOT equally likely:

Consider a bike race with multiple competitors. The outcomes (who wins) are not equally likely because some riders have more talent, better training, or superior equipment. A skilled rider has a greater chance of winning than a less experienced competitor.

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Understanding whether outcomes are equally likely is important because the basic probability formula $P(E) = \frac{n(E)}{n(S)}$ assumes that all outcomes are equally likely.

When outcomes are not equally likely, you may need to use different methods to calculate probability.

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Key Points to Remember:

Probability formula: $P(E) = \frac{n(E)}{n(S)}$ where $n(E)$ is the number of favourable outcomes and $n(S)$ is the total number of outcomes
Probability can be expressed three ways: as a fraction, decimal, or percentage - be prepared to convert between these forms
Playing cards structure: A standard deck has 52 cards in 4 suits, with 13 cards per suit and 3 picture cards per suit
Equally likely outcomes: The basic probability formula only applies when all outcomes have an equal chance of occurring
Check your answers: Probabilities must always be between 0 and 1 (or between 0% and 100%)

Definition of Probability (HSC SSCE Mathematics Standard): Revision Notes