What is the difference between an 'outcome' and an 'event'?

An outcome is a single specific result of a trial, such as rolling a 4 on a die. An event is a broader category that can include one or more outcomes, such as rolling an even number (which includes outcomes 2, 4, and 6).

How does theoretical probability differ from relative frequency?

Theoretical probability is calculated based on logic and known possibilities before any action is taken. Relative frequency is an estimate calculated after an experiment by looking at the actual results of trials.

When should you use the addition rule $P(A) + P(B)$?

This rule is used to find the probability of either event A or event B occurring, provided that the events are mutually exclusive. This means they cannot both happen at the same time during a single trial.

What is a common mistake when calculating the probability of a 'not' event?

A common error is forgetting that the sum of all probabilities must be 1, leading students to guess a value rather than subtracting the known probability from 1. Another error is failing to account for all possible outcomes that make up the 'not' condition.

Why is it incorrect to say a probability is 120%?

Probabilities are bounded by the scale of 0 to 1 (or 0% to 100%). A value over 100% is impossible because you cannot have more certainty than 'certain' or more outcomes than the total sample space allows.

What error occurs if relative frequency is calculated with too few trials?

With a small number of trials, the results are highly susceptible to random chance and may not reflect the true likelihood. This leads to an unreliable estimate that might suggest bias where none exists.

Define a 'Sample Space'.

A sample space is the set of all possible outcomes of an experiment. It defines the boundaries of what can happen, ensuring that the sum of probabilities of all items within it equals 1.

What is the formula for Theoretical Probability?

The formula is $P(A) = \frac{n(A)}{n(S)}$, where $n(A)$ is the number of successful outcomes for event A and $n(S)$ is the total number of possible outcomes in the sample space.

What are 'Mutually Exclusive' events?

These are events that cannot happen simultaneously in a single trial. For example, in a single coin flip, landing on 'Heads' and 'Tails' are mutually exclusive events.

How do you calculate 'Expected Frequency'?

You multiply the probability of the event occurring by the total number of trials planned. This gives a theoretical prediction of how many times the event should happen in that specific number of attempts.

Library Podcasts

Courses

Referral & Rewards

Maths And Numeracy Double Award / Higher

Statistics & Probability

Basic Probability

Summary

Basic probability is the mathematical study of likelihood, ranging from impossible to certain. It provides a framework for quantifying uncertainty using theoretical calculations based on known outcomes or experimental estimates derived from observed data.

1. Definition & Core Concepts

Probability describes the numerical likelihood of a specific outcome occurring within a defined set of possibilities.

An experiment is a repeatable process, such as rolling a die, where each individual repetition is referred to as a trial.

An outcome is a single possible result of a trial, while an event can be a single outcome or a collection of multiple outcomes.

The sample space represents the exhaustive list of all possible outcomes for a given experiment, often displayed as a list or a table.

A probability scale showing values from 0 to 1, with labels for impossible, unlikely, even chance, likely, and certain.

2. Underlying Principles

The Probability Scale is strictly bounded between $0$ and $1$ , where $0$ indicates an event cannot happen and $1$ indicates it must happen.

The sum of probabilities for all possible mutually exclusive outcomes in a sample space must always equal exactly $1$ .

Complementary Events are pairs where one event is the exact opposite of the other; the probability of an event not happening is calculated as $P(\text{not } A) = 1 - P(A)$ .

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips