Probability - Mathematics - Pearson A-Level

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Welcome to Probability: Understanding Uncertainty

Hello! Probability might sound intimidating with all the symbols, but at its heart, it’s just the mathematics of chance. In this chapter, we learn how to quantify the likelihood of something happening—from rolling a six on a dice to predicting weather patterns.

Why is this important? Probability forms the foundation of all advanced statistics (like hypothesis testing, which you will meet later!). Mastering these basic rules and concepts in Unit S1 is crucial for success throughout your A Level journey.

Section 1: The Core Foundations of Probability

1.1 Key Terminology

Before calculating anything, we need to speak the same statistical language:

Experiment: An action that results in an outcome (e.g., flipping a coin).
Outcome: A possible result of the experiment (e.g., Heads or Tails).
Sample Space ($S$): The list of all possible outcomes.
Example: Rolling a standard die, $S = \{1, 2, 3, 4, 5, 6\}$.
Event (A, B, C...): A specific collection of outcomes we are interested in.
Example: Event A is "rolling an even number." $A = \{2, 4, 6\}$.
Probability $P(A)$: A measure of the likelihood of Event A occurring. It must always be between 0 (impossible) and 1 (certain).

The basic formula for calculating probability (when outcomes are equally likely):

$$P(A) = \frac{\text{Number of outcomes in A}}{\text{Total number of outcomes in the Sample Space}}$$

1.2 Visualizing Events: Set Notation and Venn Diagrams

We use special symbols (Set Notation) to describe how different events relate to each other. Don't worry if these symbols seem tricky at first—just think of them as mathematical shorthand!

Key Set Notation Symbols:

Intersection ($A \cap B$): This means "A AND B." The event where both A and B occur simultaneously. Think: The shared area in a Venn Diagram.
Union ($A \cup B$): This means "A OR B." The event where A occurs, B occurs, or both occur. Think: The total area covered by circles A and B.
Complement ($A'$): This means "NOT A." The event where A does not occur. Think: Everything outside circle A, but still within the Sample Space (S).
Empty Set ($\emptyset$): Represents an impossible event. $P(\emptyset) = 0$.

A Venn Diagram is the best way to visualize these relationships. The rectangle represents the Sample Space $S$, and the circles represent the events A and B.

Quick Review: Probability is always a fraction or decimal between 0 and 1. The symbols $\cap$ and $\cup$ represent "AND" and "OR" respectively.

Section 2: The Rules of Probability

2.1 The Complement Rule (What if it doesn't happen?)

Since the probability of something happening plus the probability of it not happening must equal 1 (certainty), we get the Complement Rule:

$$P(A') = 1 - P(A)$$

Memory Aid: Use this rule when calculating $P(A)$ is complicated, but calculating $P(A')$ is easier. For example, finding the probability of "at least one six" when rolling three dice is much easier by calculating $1 - P(\text{no sixes})$.

2.2 The Addition Rule (Handling "OR" events)

When you want to find the probability of Event A OR Event B occurring ($P(A \cup B)$), you must consider whether the events overlap.

The General Addition Rule:

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Why the subtraction? If A and B overlap (i.e., $P(A \cap B) > 0$), you included the overlap area twice—once when counting A, and once when counting B. You must subtract $P(A \cap B)$ once to count it correctly.

2.3 Mutually Exclusive Events

Definition: Two events are mutually exclusive if they cannot happen at the same time. They have absolutely no outcomes in common.

If A and B are mutually exclusive, their intersection is the empty set: $A \cap B = \emptyset$.
Therefore, the probability of their intersection is zero: $P(A \cap B) = 0$.

Special Addition Rule (for Mutually Exclusive Events):

$$P(A \cup B) = P(A) + P(B)$$ (Since $P(A \cap B)$ is zero, we don't need to subtract anything!)

Example: You cannot roll a 1 (Event A) AND a 6 (Event B) simultaneously on a single standard die roll. A and B are mutually exclusive.

Common Mistake Alert! Do NOT confuse Mutually Exclusive with Independent (Section 3). They are almost opposites!

Mutually Exclusive: Events prevent each other from happening.
Independent: Events have no influence on each other.

Key Takeaway: Use the Addition Rule for "OR" questions. Always check if the events are Mutually Exclusive—if they are, the formula simplifies significantly!

Section 3: Conditional Probability and Independence

3.1 Understanding Conditional Probability

Sometimes, knowing that one event has occurred changes the probability of a second event. This is Conditional Probability.

Definition: The probability of Event A occurring GIVEN that Event B has already occurred is written as $P(A|B)$.

Analogy: Imagine you are finding the probability that a person likes chocolate. If you are told that person is already a child (the condition B), that probability might change, as children often have a higher preference for chocolate than adults. The condition $B$ shrinks your sample space.

The Formula for Conditional Probability:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

(This formula holds true, provided that $P(B) \neq 0$.)

3.2 The Multiplication Rule (Handling "AND" events)

We can rearrange the Conditional Probability formula to find the probability of both A AND B happening:

The General Multiplication Rule:

$$P(A \cap B) = P(A) \times P(B|A) \quad \text{OR} \quad P(A \cap B) = P(B) \times P(A|B)$$

This rule is essential when dealing with sequences of dependent events (like picking two socks from a drawer without putting the first one back).

3.3 Independent Events

Definition: Two events A and B are independent if the occurrence of one does not affect the probability of the other.

If A and B are independent, knowing B occurred doesn't change the probability of A, meaning $P(A|B) = P(A)$.

Test for Independence: If A and B are independent, the Multiplication Rule simplifies dramatically:

$$P(A \cap B) = P(A) \times P(B)$$

How to use the Independence Test: 1. Calculate $P(A \cap B)$ from the data (e.g., using a Venn diagram or table). 2. Calculate $P(A) \times P(B)$. 3. If the results from steps 1 and 2 are equal, the events are independent.

Did you know? Independence is crucial in quality control. If the chance of one item being defective is independent of the previous one, we can calculate the probability of two defects very easily!

Key Takeaway: Conditional probability $P(A|B)$ is about finding probability within a reduced sample space. Use the simplified multiplication rule $P(A \cap B) = P(A)P(B)$ only if you KNOW or are testing for independence.

Section 4: Using Tree Diagrams

Tree Diagrams are fantastic visual tools for representing sequences of events, especially when the events are dependent (i.e., conditional probability is involved).

4.1 Constructing a Tree Diagram

Follow these steps:

Start Node: The starting point represents the start of the experiment.
First Event Branches: Draw branches for all outcomes of the first event (e.g., Rain or No Rain). Write the probability on each branch.
Second Event Branches: From the end of each first branch, draw new branches for the second event. Crucially: These probabilities are often conditional, based on the path taken so far.
List Outcomes: At the very end of the tree, list the final combined outcomes (e.g., R, R' or R, C).
Calculate Path Probabilities: To find the probability of a specific sequence, multiply the probabilities along that path.

$$P(\text{Path}) = P(\text{Branch 1}) \times P(\text{Branch 2|Branch 1})$$

4.2 Dependent Events (Without Replacement)

Tree diagrams are most helpful when dealing with events without replacement (where the total sample space changes after the first pick).

Example: A bag contains 5 red balls and 5 blue balls (Total 10). You pick two balls without replacement.

First Pick: $$P(\text{R1}) = 5/10$$ $$P(\text{B1}) = 5/10$$

Second Pick (Conditional): If you picked Red first (R1), there are now 4 red and 5 blue balls left (Total 9). $$P(\text{R2}|\text{R1}) = 4/9$$ $$P(\text{B2}|\text{R1}) = 5/9$$

To find the probability of two reds: $$P(\text{R1} \cap \text{R2}) = P(\text{R1}) \times P(\text{R2}|\text{R1}) = \frac{5}{10} \times \frac{4}{9} = \frac{20}{90}$$

Tip for Tree Diagrams: To solve an "OR" problem (e.g., finding the probability of getting one red AND one blue), calculate the probabilities of ALL successful paths and add them together.

Key Takeaway: Tree diagrams use the multiplication rule for sequences (along the branches) and the addition rule for combining successful paths (at the end). Be meticulous when adjusting probabilities for "without replacement" scenarios!