Probability

Welcome to Probability for AS 2 Applied Mathematics

Welcome to one of the most practical and exciting areas of mathematics! Whether predicting the weather, assessing risk in finance, or testing new medical treatments, probability is the mathematical tool we use to measure uncertainty. Don't worry if probability has felt confusing in the past. In this chapter, we will break down every single rule into simple, logical steps with visual diagrams, clear formulas, and relatable real-world examples.

What you will master in this chapter:
• Fundamental probability rules and notation
• The Complement Rule and working with sample spaces
• Venn diagrams, Two-Way tables, and Tree diagrams
• The Addition Rule and Mutually Exclusive events
• Conditional Probability and the Multiplication Rule
• Independent events and how to test for independence in exams

1. Fundamental Ideas and Notation

Sample Space and Probability Scale

In statistics, an experiment is any repeatable process that yields an outcome (like rolling a fair die). The set of all possible outcomes is called the sample space, often denoted by the symbol \(S\) or \(\mathcal{E}\). An event is a collection of one or more outcomes that we are interested in (for example, rolling an even number).

For any event \(A\), its probability \(P(A)\) must lie between \(0\) and \(1\) inclusive:
\(0 \le P(A) \le 1\)
• If \(P(A) = 0\), the event is completely impossible.
• If \(P(A) = 1\), the event is completely certain.
• The sum of the probabilities of all mutually exclusive, exhaustive outcomes in the sample space always equals \(1\): \(\sum P = 1\).

The Complement of an Event

The complement of event \(A\) is the event that "\(A\) does not happen". We write this as \(A'\) (or sometimes \(A^c\) or \(\bar{A}\)).

Because an event either happens or does not happen, their probabilities must add up to \(1\):
\(P(A) + P(A') = 1 \implies P(A') = 1 - P(A)\)

Example: If the probability that it rains in Belfast tomorrow is \(P(\text{Rain}) = 0.35\), then the probability that it does not rain is:
\(P(\text{Rain}') = 1 - 0.35 = 0.65\)

Quick Review: Every probability is between \(0\) and \(1\). Whenever you see the word "not", immediately think of the complement rule: \(P(\text{not } A) = 1 - P(A)\).

2. Combining Events: Unions, Intersections, and Venn Diagrams

Understanding the Language and Symbols

When dealing with two events \(A\) and \(B\), we often look at combinations of these events:

1. Intersection: "\(A\) AND \(B\)" (\(A \cap B\))
• The symbol \(\cap\) looks like an arch or an "n" for and.
• \(A \cap B\) means both event \(A\) and event \(B\) occur at the same time.
• On a Venn diagram, this is the overlapping region where the two circles cross.

2. Union: "\(A\) OR \(B\)" (\(A \cup B\))
• The symbol \(\cup\) looks like a cup or a "u" for union.
• \(A \cup B\) means event \(A\) occurs, event \(B\) occurs, or both occur.
• On a Venn diagram, this represents everything inside circle \(A\), circle \(B\), or their shared middle.

The General Addition Rule

If you add all elements in circle \(A\) to all elements in circle \(B\), you count the middle overlapping section twice! To fix this double-counting, we subtract the intersection once.

The General Addition Formula:
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

You can rearrange this formula depending on what the question asks for:
\(P(A \cap B) = P(A) + P(B) - P(A \cup B)\)

Mutually Exclusive Events

Two events \(A\) and \(B\) are mutually exclusive if they cannot happen at the same time. They have zero overlap.
Everyday analogy: You cannot turn left and turn right at the exact same moment.

For mutually exclusive events:
• \(P(A \cap B) = 0\)
• On a Venn diagram, the circles do not touch or overlap at all.
• The addition rule simplifies to: \(P(A \cup B) = P(A) + P(B)\)

Worked Example:
In a sixth-form group of \(100\) students, \(60\) study Mathematics (\(M\)), \(40\) study Physics (\(P\)), and \(25\) study both. A student is chosen at random.
Find the probability that the student studies Mathematics or Physics.
• \(P(M) = \frac{60}{100} = 0.60\)
• \(P(P) = \frac{40}{100} = 0.40\)
• \(P(M \cap P) = \frac{25}{100} = 0.25\)
Using the addition rule:
\(P(M \cup P) = P(M) + P(P) - P(M \cap P) = 0.60 + 0.40 - 0.25 = 0.75\)

Key Takeaway: Whenever you see "or", use the addition rule. Always remember to check if you need to subtract \(P(A \cap B)\).

3. Conditional Probability

What is Conditional Probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It restricts our focus to a smaller sub-group of the sample space.

We write "\(P(A|B)\)", which is read aloud as "the probability of \(A\) given \(B\)".
The vertical bar "|" means "given that". The condition that comes after the bar is the condition we already know to be true.

The Conditional Probability Formula

Because we know event \(B\) has definitely happened, \(B\) becomes our new total universe. Out of this universe \(B\), we want to know the portion where \(A\) also happens (\(A \cap B\)).

Formula:
\(P(A|B) = \frac{P(A \cap B)}{P(B)}\), provided \(P(B) > 0\)

Similarly:
\(P(B|A) = \frac{P(A \cap B)}{P(A)}\), provided \(P(A) > 0\)

The Multiplication Rule

By rearranging the conditional probability formula, we get the Multiplication Rule:
\(P(A \cap B) = P(A|B) \times P(B)\)
or
\(P(A \cap B) = P(B|A) \times P(A)\)

Working with Two-Way Tables

Two-way contingency tables make conditional probability very easy to visualize because you can simply read the numbers directly from the restricted row or column.

Example Table:
Consider \(100\) students surveyed about whether they play a musical instrument:

• Year 13: Instrument = \(30\), No Instrument = \(20\), Total = \(50\)
• Year 14: Instrument = \(15\), No Instrument = \(35\), Total = \(50\)
• Overall Totals: Instrument = \(45\), No Instrument = \(55\), Total = \(100\)

Let \(I\) be the event "plays an instrument" and \(Y_{13}\) be the event "in Year 13".
1. Find \(P(I)\):
\(P(I) = \frac{45}{100} = 0.45\)
2. Find \(P(I | Y_{13})\):
Method: Look only at the Year 13 row (total is \(50\)). Out of these \(50\), \(30\) play an instrument.
\(P(I | Y_{13}) = \frac{30}{50} = 0.60\)
Using the formula to verify: \(P(I | Y_{13}) = \frac{P(I \cap Y_{13})}{P(Y_{13})} = \frac{30/100}{50/100} = \frac{30}{50} = 0.60\).

Key Takeaway: The term after the vertical bar "|" always goes in the denominator!

4. Independent Events

Definition of Independence

Two events \(A\) and \(B\) are independent if the occurrence of one event has no effect whatsoever on the probability of the other event occurring.
Everyday analogy: Tossing "Heads" on a coin does not change the probability that it will rain outside.

Mathematically, if \(A\) and \(B\) are independent:
\(P(A|B) = P(A)\) and \(P(B|A) = P(B)\)

The Test for Independence

Substituting \(P(A|B) = P(A)\) into the multiplication rule gives the most important formula for independent events:

Multiplication Rule for Independent Events:
\(P(A \cap B) = P(A) \times P(B)\)

Exam Tip - How to Prove Independence:
In CCEA exam questions, you are often asked: "Determine whether events \(A\) and \(B\) are statistically independent."
Follow these three simple steps:
1. Calculate the value of \(P(A \cap B)\) from the given data or diagram.
2. Calculate the separate product: \(P(A) \times P(B)\).
3. Compare the two numbers:
• If \(P(A \cap B) = P(A) \times P(B)\), state: "Since \(P(A \cap B) = P(A) \times P(B)\), the events are independent."
• If \(P(A \cap B) \ne P(A) \times P(B)\), state: "Since \(P(A \cap B) \ne P(A) \times P(B)\), the events are not independent."

Common Confusion: Mutually Exclusive vs. Independent

Students often mix these two terms up, but they mean completely different things!

• Mutually Exclusive: Events cannot happen together \(\implies P(A \cap B) = 0\). (They are dependent, because if one happens, the other definitely cannot happen!)
• Independent: Events can happen together, but do not influence each other \(\implies P(A \cap B) = P(A) \times P(B)\).

Did you know? If two events \(A\) and \(B\) both have non-zero probabilities (\(P(A) > 0\) and \(P(B) > 0\)), they cannot be both mutually exclusive and independent at the same time!

5. Tree Diagrams and Sequential Events

Tree diagrams are ideal for representing sequences of events occurring one after another, especially when dealing with problems involving selection with replacement (independent trials) or without replacement (conditional trials).

The Golden Rules of Tree Diagrams

1. Along the branches: We multiply probabilities to find the intersection of those specific outcomes.
\(P(\text{Path}) = P(\text{Branch 1}) \times P(\text{Branch 2})\)
2. Down the tree: We add the probabilities of distinct end-paths that satisfy the required condition.
3. At each node: The branches coming out of any single point must always sum to \(1\).

Step-by-Step Example (Without Replacement)

A bag contains \(5\) Red balls and \(3\) Blue balls (total \(8\) balls). Two balls are drawn at random without replacement.
Find the probability of picking exactly one Red ball.

Step 1: First Selection
• Probability of Red (\(R_1\)) = \(\frac{5}{8}\)
• Probability of Blue (\(B_1\)) = \(\frac{3}{8}\)

Step 2: Second Selection (One ball is now gone, so total is \(7\))
• If first was Red: \(P(R_2 | R_1) = \frac{4}{7}\), and \(P(B_2 | R_1) = \frac{3}{7}\)
• If first was Blue: \(P(R_2 | B_1) = \frac{5}{7}\), and \(P(B_2 | B_1) = \frac{2}{7}\)

Step 3: Identify Paths with "Exactly One Red"
There are two successful paths: (Red then Blue) OR (Blue then Red).
• Path 1 (\(R_1 \cap B_2\)): \(\frac{5}{8} \times \frac{3}{7} = \frac{15}{56}\)
• Path 2 (\(B_1 \cap R_2\)): \(\frac{3}{8} \times \frac{5}{7} = \frac{15}{56}\)

Step 4: Add the Paths
\(P(\text{Exactly one Red}) = \frac{15}{56} + \frac{15}{56} = \frac{30}{56} = \frac{15}{28}\)

Key Takeaway: When objects are chosen without replacement, remember to decrease both the numerator and the denominator for conditional probabilities on second-stage branches!

6. Summary of Key Formulas & Exam Checklist

Formula Sheet Summary

• Complement: \(P(A') = 1 - P(A)\)
• General Addition Rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
• Mutually Exclusive Events: \(P(A \cap B) = 0 \implies P(A \cup B) = P(A) + P(B)\)
• Conditional Probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
• General Multiplication Rule: \(P(A \cap B) = P(A|B) \times P(B)\)
• Independent Events: \(P(A|B) = P(A) \iff P(A \cap B) = P(A) \times P(B)\)

Top Common Mistakes to Avoid in Exams

• Forgetting the denominator in conditional probability: Remember that \(P(A|B)\) has \(P(B)\) on the bottom, not \(P(A)\).
• Assuming independence without testing: Never use \(P(A \cap B) = P(A) \times P(B)\) unless the question explicitly states the events are independent or you are testing for it.
• Forgetting to subtract the intersection in Venn diagrams: When filling in a circle for \(P(A)\), remember the entire circle sums to \(P(A)\); the "only \(A\)" region is \(P(A) - P(A \cap B)\).
• Not showing formal numerical comparison for independence: Always calculate \(P(A \cap B)\) and \(P(A) \times P(B)\) separately, show their values clearly, and write a concluding sentence.