Probability

Probability Basics

Year 7 · Year 8 · Year 9

✓Understand and use the probability scale from 0 to 1.
✓Calculate the probability of a single event using the formula.
✓List all possible outcomes of an event using sample space diagrams.
✓Use appropriate probability notation and terminology.

Key concepts

Probability

Probability is a measure of how likely an event is to happen. It is expressed as a number between 0 and 1.

Probability Scale

The probability scale ranges from 0 to 1. An event with a probability of 0 is impossible, an event with a probability of 1 is certain. A probability of 0.5 (or 1/2) means an event has an even chance of happening. Probabilities can be written as fractions, decimals, or percentages.

Event

An event is a specific outcome or a set of outcomes that we are interested in.

Outcome

An outcome is one of the possible results of an experiment or situation.

Sample Space

The sample space is the set of all possible outcomes of an experiment. It is often represented using a list, table, or tree diagram.

Calculating Probability of a Single Event

The probability of a single event, P(Event), is calculated by dividing the number of favourable outcomes (the outcomes we are interested in) by the total number of possible outcomes.

P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Complementary Events

Two events are complementary if they are the only two possible outcomes and one must happen. The sum of the probabilities of an event and its complement is always 1. For example, P(A) + P(not A) = 1.

P(A) + P(not A) = 1

Key facts to remember

1Probability measures the likelihood of an event occurring.
2The probability of any event is always between 0 and 1, inclusive (0 ≤ P(Event) ≤ 1).
3P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes).
4A probability of 0 means an event is impossible; a probability of 1 means an event is certain.
5Probabilities can be expressed as fractions, decimals, or percentages.
6The sum of the probabilities of all possible outcomes in a sample space is 1.
7For complementary events, P(A) + P(not A) = 1.
8A sample space diagram is a systematic way to list all possible outcomes of an experiment.

Worked examples

Example 1

A standard six-sided die is rolled. a) What is the probability of rolling a 7? b) What is the probability of rolling a number less than 7? c) What is the probability of rolling an even number?

IIdentify the total number of possible outcomes when rolling a standard six-sided die: {1, 2, 3, 4, 5, 6}. So, Total outcomes = 6.

IIa) The number 7 is not on a standard six-sided die. Number of favourable outcomes = 0.

IIIP(rolling a 7) = 0/6 = 0.

IVb) Numbers less than 7 are {1, 2, 3, 4, 5, 6}. Number of favourable outcomes = 6.

VP(rolling a number less than 7) = 6/6 = 1.

VIc) Even numbers are {2, 4, 6}. Number of favourable outcomes = 3.

VIIP(rolling an even number) = 3/6 = 1/2.

Answer

a) P(rolling a 7) = 0 b) P(rolling a number less than 7) = 1 c) P(rolling an even number) = 1/2

Always simplify fractions to their lowest terms.

Example 2

A bag contains 5 red counters, 3 blue counters, and 2 green counters. A counter is chosen at random from the bag. a) What is the probability that the counter is red? b) What is the probability that the counter is not blue?

ICalculate the total number of counters in the bag: 5 + 3 + 2 = 10 counters.

IIa) Number of red counters = 5.

IIIP(red) = (Number of red counters) / (Total number of counters) = 5/10.

IVSimplify the fraction: 5/10 = 1/2.

Vb) Number of blue counters = 3. Number of counters that are not blue = Total counters - Number of blue counters = 10 - 3 = 7.

VIAlternatively, not blue means red or green: 5 + 2 = 7.

VIIP(not blue) = (Number of not blue counters) / (Total number of counters) = 7/10.

Answer

a) P(red) = 1/2 b) P(not blue) = 7/10

Remember that P(not A) = 1 - P(A).

Example 3

Two fair coins are tossed. a) List all the possible outcomes using a sample space diagram. b) What is the probability of getting two heads? c) What is the probability of getting at least one tail?

Ia) Create a table or list to show all combinations for Coin 1 and Coin 2.

IICoin 1 can be Head (H) or Tail (T).

IIICoin 2 can be Head (H) or Tail (T).

IVPossible outcomes: (H, H), (H, T), (T, H), (T, T).

VTotal number of outcomes = 4.

VIb) Identify the outcome for two heads: (H, H). Number of favourable outcomes = 1.

VIIP(two heads) = 1/4.

VIIIc) Identify outcomes with at least one tail: (H, T), (T, H), (T, T). Number of favourable outcomes = 3.

9P(at least one tail) = 3/4.

Answer

a) Sample space: {(H, H), (H, T), (T, H), (T, T)} b) P(two heads) = 1/4 c) P(at least one tail) = 3/4

A sample space diagram helps to ensure all possible outcomes are listed systematically.

Common mistakes

✗Stating a probability as a number less than 0 or greater than 1.
✗Not simplifying fractions to their lowest terms (e.g., leaving 5/10 instead of 1/2).
✗Incorrectly identifying the total number of possible outcomes.
✗Incorrectly identifying the number of favourable outcomes for an event.
✗Missing outcomes when creating a sample space diagram, especially for multiple events.

Exam tips

★Always simplify your probability fractions to their lowest terms unless otherwise specified.
★Clearly show your working, including the formula used, especially for multi-step problems.
★When asked to list outcomes, be systematic (e.g., use a table or an ordered list) to ensure you don't miss any.
★Double-check that your final probability answer is between 0 and 1.
★Understand the difference between 'at least', 'at most', and 'exactly' when counting favourable outcomes.

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