What is the key difference between an **outcome** and an **event** in probability?

An **outcome** is a single, specific result of an experiment (e.g., rolling a 3). An **event** is a collection of one or more outcomes (e.g., rolling an odd number, which includes 1, 3, and 5). Understanding this distinction helps in correctly identifying what constitutes a "favorable" result.

How do **mutually exclusive events** differ from **complementary events**?

**Mutually exclusive events** are any two events that cannot occur at the same time. **Complementary events** are a special type of mutually exclusive events where one event *must* occur if the other does not, and together they cover the entire sample space (e.g., rain or no rain). All complementary events are mutually exclusive, but not all mutually exclusive events are complementary.

When is an experiment considered **fair**, and how does this contrast with a **biased** experiment?

An experiment is **fair** if all its possible outcomes have an equal chance of occurring, such as a standard coin flip or a fair die roll. A **biased** experiment, conversely, has outcomes where some are more likely than others, like a weighted coin. The assumption of fairness is critical for applying basic theoretical probability formulas.

What common error occurs if you forget that the sum of all probabilities in a sample space must equal 1?

Forgetting this rule can lead to incorrect calculations for missing probabilities or an inability to verify the consistency of a probability distribution. If the sum is not 1, it indicates an error in identifying all outcomes or in their assigned probabilities.

What is a common mistake when calculating $P(A \text{ or } B)$ for events that are *not* mutually exclusive?

A common mistake is to simply add $P(A) + P(B)$ even when the events are not mutually exclusive. This leads to double-counting the probability of outcomes that are common to both events. For non-mutually exclusive events, the correct formula would involve subtracting the probability of their intersection.

What is the pitfall of using a theoretical probability formula when outcomes are *not* equally likely?

Using the formula $P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$ assumes all outcomes are equally likely. If outcomes are not equally likely (e.g., a biased coin), this formula will yield an incorrect probability, as it doesn't account for the varying likelihoods of individual outcomes.

Define **sample space** in the context of probability.

The **sample space** is the complete set of all possible individual outcomes of a random experiment. It serves as the foundational set from which events are defined and probabilities are calculated, ensuring all potential results are considered.

What is a **complementary event**?

A **complementary event** to event A, denoted as $A'$, is the event that A does not occur. The probability of a complementary event is always $1 - P(A)$, meaning that A and A' together cover all possibilities in the sample space.

What does it mean for two events to be **mutually exclusive**?

Two events are **mutually exclusive** if they cannot happen at the same time during a single trial of an experiment. Their occurrences are entirely separate, meaning there is no overlap between their sets of outcomes.

State the formula for calculating the **theoretical probability** of an event A, assuming equally likely outcomes.

The theoretical probability of event A is given by $P(A) = \frac{\text{Number of outcomes favorable to A}}{\text{Total number of possible outcomes}}$. This formula is applicable when each outcome in the sample space has an identical chance of occurring.

Basic Probability | Pearson Edexcel IGCSE Maths

A Modular / Higher Unit 1

Statistics & Probability

Basic Probability

Summary

Basic probability quantifies the likelihood of events occurring, using a numerical scale from 0 (impossible) to 1 (certain). It relies on fundamental concepts like outcomes, events, and sample spaces, and provides methods for calculating theoretical probabilities, understanding complementary events, and combining probabilities for mutually exclusive events. This foundational understanding is crucial for more advanced statistical analysis and decision-making under uncertainty.

1. Introduction to Probability

Definition: Probability is a numerical measure of the likelihood that a specific event will occur. It provides a quantitative way to express uncertainty, ranging from events that are impossible to those that are certain.
Probability Scale: Probabilities are expressed as a number between 0 and 1, inclusive. A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event. Values between 0 and 0.5 represent unlikely events, 0.5 signifies an even chance, and values between 0.5 and 1 denote likely events.
Representation: Probabilities can be presented as fractions, decimals, or percentages. For instance, an even chance can be expressed as $\frac{1}{2}$ , $0.5$ , or $50\%$ , depending on the context or required format. Using fractions often maintains precision and avoids rounding errors.

2. Fundamental Terminology in Probability

Experiment: An experiment is any process or activity that can be repeated and has a set of well-defined possible results. Examples include rolling a die, flipping a coin, or drawing a card from a deck.
Trial: Each individual performance of an experiment is called a trial. For example, if a coin is flipped 10 times, each flip constitutes a single trial.
Outcome: An outcome is a single possible result of a trial. When rolling a standard six-sided die, the outcomes are 1, 2, 3, 4, 5, or 6.
Event: An event is a specific outcome or a collection of outcomes from an experiment. For example, rolling an even number on a die is an event that includes the outcomes {2, 4, 6}. Events are often denoted by capital letters, such as Event A.
Sample Space: The sample space is the complete set of all possible outcomes of an experiment. It is typically represented by a list or a set notation, ensuring all potential results are accounted for. For a single coin flip, the sample space is {Heads, Tails}.
Fair vs. Biased Events: An event is considered fair if every outcome in the sample space has an equal chance of occurring. Conversely, an event is biased if some outcomes are more likely than others, such as a weighted coin or a loaded die.

3. Calculating Theoretical Probability

4. Properties of Probability

5. Mutually Exclusive Events

6. Key Distinctions and Relationships

7. Exam Strategy & Common Pitfalls