Chapter 9: Probability of Combined Events

Overview

Welcome to Chapter 9 of Form 4 Mathematics! This chapter introduces you to the exciting world of probability and combined events. You'll learn to calculate probabilities for various types of combined events, understand the differences between independent and dependent events, and master the use of tree diagrams for complex probability calculations. Probability is essential for understanding uncertainty and making informed decisions.

What You'll Learn:

• Understand and describe combined events
• Distinguish between dependent and independent events
• Apply probability formulas for combined events
• Use tree diagrams for complex probability calculations

Learning Objectives

After completing this chapter, you will be able to:

• Describe combined events and list possible outcomes
• Distinguish between dependent and independent events
• Verify and use probability formulas for combined events
• Use tree diagrams to solve probability problems

Probability Fundamentals

Basic Probability Formula

The probability of an event A is given by:

$$P(A) = \frac{n(A)}{n(S)}$$

Where:

• $n(A)$ = number of favorable outcomes for event A
• $n(S)$ = total number of outcomes in the sample space

Probability Rules

1. Range of Probability:

$$0 \leq P(A) \leq 1$$

2. Complement Rule:

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

3. Addition for Mutually Exclusive Events:

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

4. Multiplication for Independent Events:

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

5. Conditional Probability:

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

Key Concepts

Combined Events

Combined events are events that involve two or more events occurring together. Outcomes can be represented using ordered pairs, tree diagrams, or tables.

Types of Combined Events:

• Joint Events: Both events occur simultaneously
• Compound Events: One event occurs given another has occurred
• Union Events: At least one of the events occurs

Example: Rolling two dice: Combined events include (1,1), (1,2), (1,3), ..., (6,6)

Sample Space Visualization

For rolling two dice, the sample space can be visualized as a table:

Event Types Diagram

Venn Diagram for Event Relationships

Set Notation:

• $A \cap B$ = Intersection (both events occur)
• $A \cup B$ = Union (at least one event occurs)
• $A'$ = Complement (event A does not occur)

Independent vs. Dependent Events

Independent Events:

• Two events where the occurrence of the first does not affect the occurrence of the second
• P(A and B) = P(A) × P(B)

Example:

• Rolling a die twice
• Drawing cards with replacement

Dependent Events:

• Two events where the occurrence of the first affects the occurrence of the second
• P(A and B) = P(A) × P(B|A)

Example:

• Drawing cards without replacement
• Selecting items without replacement

Mutually Exclusive vs. Non-Mutually Exclusive Events

Mutually Exclusive Events:

• Two events that cannot occur at the same time
• P(A ∩ B) = ∅ (empty set)
• P(A or B) = P(A) + P(B)

Example:

• Getting heads and tails on the same coin flip
• Being born in January and February

Non-Mutually Exclusive Events:

• Two events that can occur at the same time
• P(A ∩ B) ≠ ∅
• P(A or B) = P(A) + P(B) - P(A ∩ B)

Example:

• Drawing a red card or a king from a deck
• Being a student and working part-time

Important Formulas and Methods

Probability Rules for Combined Events

1. Multiplication Rule: For two events A and B:

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

If A and B are independent:

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

2. Addition Rule: For mutually exclusive events:

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

For non-mutually exclusive events:

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

3. Conditional Probability:

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

Tree Diagrams

Tree diagrams are visual tools that list all possible outcomes and help calculate probabilities for combined events.

Structure:

• First branch: First event with probabilities
• Second branch: Second event with probabilities
• End nodes: Final outcomes with joint probabilities

Example for two coin flips:

        H (0.5)
    /   \
   /     \
H (0.5)   T (0.5)
   \     /
    \   /
     T (0.5)

Joint probabilities:

• P(HH) = 0.5 × 0.5 = 0.25
• P(HT) = 0.5 × 0.5 = 0.25
• P(TH) = 0.5 × 0.5 = 0.25
• P(TT) = 0.5 × 0.5 = 0.25

Advanced Tree Diagram Examples

Example: Three Coin Flips

Example: Card Drawing with Replacement

Probability Flow Chart

Bayes' Theorem Application

For conditional probability in reverse:

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

Medical Testing Example:

• P(Disease) = 0.01 (1% prevalence)
• P(Positive|Disease) = 0.95 (95% true positive rate)
• P(Positive|No Disease) = 0.05 (5% false positive rate)

Step-by-Step Solved Examples

Example 1: Independent Events

Problem: A bag contains 3 red and 2 blue marbles. A marble is drawn, replaced, and then another marble is drawn. Find the probability of drawing two red marbles.

Solution: Event A: First marble is red Event B:** Second marble is red

Since replacement is used, the events are independent.

P(A) = 3/5 (3 red out of 5 total) P(B) = 3/5 (same probability due to replacement)

P(A and B) = P(A) × P(B) = (3/5) × (3/5) = 9/25

Answer: Probability of two red marbles is 9/25

Example 2: Dependent Events

Problem: A bag contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. Find the probability of drawing two red marbles.

Solution: Event A: First marble is red Event B:** Second marble is red (given first was red)

Since no replacement, the events are dependent.

P(A) = 5/8 (5 red out of 8 total) P(B|A) = 4/7 (4 red left out of 7 remaining)

P(A and B) = P(A) × P(B|A) = (5/8) × (4/7) = 20/56 = 5/14

Answer: Probability of two red marbles is 5/14

Example 3: Mutually Exclusive Events

Problem: A card is drawn from a standard deck. Find the probability of drawing a king or a queen.

Solution: Event A: Drawing a king Event B:** Drawing a queen

These are mutually exclusive (a card cannot be both king and queen).

P(A) = 4/52 (4 kings out of 52 cards) P(B) = 4/52 (4 queens out of 52 cards)

P(A or B) = P(A) + P(B) = 4/52 + 4/52 = 8/52 = 2/13

Answer: Probability of drawing a king or queen is 2/13

Example 4: Non-Mutually Exclusive Events

Problem: A card is drawn from a standard deck. Find the probability of drawing a red card or a king.

Solution: Event A: Drawing a red card Event B:** Drawing a king

These are not mutually exclusive (there are red kings).

P(A) = 26/52 (26 red cards) P(B) = 4/52 (4 kings) P(A and B) = 2/52 (2 red kings)

P(A or B) = P(A) + P(B) - P(A and B) = 26/52 + 4/52 - 2/52 = 28/52 = 7/13

Answer: Probability of drawing a red card or king is 7/13

Example 7: Binomial Probability

Problem: A multiple-choice test has 10 questions, each with 4 choices. A student guesses all answers. Find the probability of getting exactly 7 correct answers.

Solution: Parameters:

• n = 10 (number of trials)
• k = 7 (number of successes)
• p = 0.25 (probability of success)
• q = 0.75 (probability of failure)

Binomial Formula:

$$P(X = k) = \binom{n}{k} p^k q^{n-k}$$

Where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Calculation: $\binom{10}{7} = \frac{10!}{7!3!} = \frac{10×9×8}{3×2×1} = 120$

P(X=7) = 120 × (0.25)⁷ × (0.75)³ = 120 × 0.000061 × 0.421875 ≈ 0.0031

Answer: Probability is approximately 0.0031 (0.31%)

Example 8: Probability in Genetics

Problem: In Mendelian genetics, if two heterozygous parents (Aa × Aa) have children, what's the probability of: a) 3 children with recessive trait b) At least 1 child with dominant trait

Solution: P(Recessive) = P(aa) = 1/4 P(Dominant) = P(AA or Aa) = 3/4

a) P(3 recessive): Since independent events: P = (1/4)³ = 1/64

b) P(at least 1 dominant): Easier to calculate complement: P(at least 1 dominant) = 1 - P(all recessive) = 1 - (1/4)³ = 1 - 1/64 = 63/64

Answer: a) 1/64, b) 63/64

Example 5: Tree Diagram Problem

Problem: A company has two machines. Machine A produces 60% of the output, and Machine B produces 40%. Machine A has a defect rate of 5%, and Machine B has a defect rate of 8%. If an item is found to be defective, what is the probability it came from Machine A?

Solution: Tree Diagram:

Machine A (0.6)
  /   \
 /     \
Good (0.95) Defective (0.05)
Machine B (0.4)
  \     /
   \   /
Good (0.92) Defective (0.08)

Calculate probabilities: P(A and Good) = 0.6 × 0.95 = 0.57 P(A and Defective) = 0.6 × 0.05 = 0.03 P(B and Good) = 0.4 × 0.92 = 0.368 P(B and Defective) = 0.4 × 0.08 = 0.032

Total defective probability: P(Defective) = 0.03 + 0.032 = 0.062

Find P(A|Defective): P(A|Defective) = P(A and Defective) / P(Defective) = 0.03 / 0.062 ≈ 0.4839

Answer: Probability it came from Machine A is approximately 0.484

Example 6: Complex Combined Events

Problem: A fair die is rolled twice. Find: a) P(sum is 7) b) P(sum is even or first roll is 1)

Solution: Sample space: 6 × 6 = 36 possible outcomes

a) P(sum is 7): Outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) P(sum=7) = 6/36 = 1/6

b) P(sum is even or first roll is 1): Event A: Sum is even (18 outcomes) Event B: First roll is 1 (6 outcomes: (1,1) to (1,6)) A and B: First roll is 1 and sum is even (3 outcomes: (1,1), (1,3), (1,5))

P(A or B) = P(A) + P(B) - P(A and B) = 18/36 + 6/36 - 3/36 = 21/36 = 7/12

Answer: a) 1/6, b) 7/12

Real-world Applications

1. Business and Finance

Risk Assessment:

Quality Control:

• Statistical Process Control (SPC): Monitor production processes using probability
• Acceptance Sampling: Determine sample sizes for quality assurance
• Defect Rate Analysis: Calculate probabilities of product failures

Example: A factory produces 10,000 units daily with a 2% defect rate. What's the probability of finding exactly 5 defective units in a random sample of 100 units?

2. Medicine and Healthcare

Diagnostic Testing:

Clinical Trials:

• Sample Size Calculation: Power analysis using probability
• Efficacy Testing: Compare treatment vs. placebo success rates
• Safety Monitoring: Probability of adverse events

3. Insurance and Risk Management

Premium Calculation:

• Life Insurance: Based on mortality tables and life expectancy probabilities
• Auto Insurance: Accident probability based on driving history, age, location
• Property Insurance: Risk assessment based on location, building type, weather patterns

Risk Pooling:

4. Games and Entertainment

Game Design:

• Slot Machines: Probability calculations for payouts and house edge
• Card Games: Understanding odds and probabilities for strategic play
• Lottery Design: Balance between jackpot size and winning probability

Sports Analytics:

5. Scientific Research

Experimental Design:

• Power Analysis: Determine sample sizes needed to detect effects
• Hypothesis Testing: Calculate p-values and confidence levels
• Regression Analysis: Predict relationships between variables

Data Analysis:

• Statistical Significance: Distinguish between real effects and random chance
• Confidence Intervals: Estimate population parameters from samples
• Bayesian Inference: Update probabilities based on new evidence

6. Engineering and Technology

Reliability Engineering:

• Failure Analysis: Calculate system reliability and MTBF (Mean Time Between Failures)
• Quality Assurance: Probability-based testing strategies
• Risk Assessment: Failure modes and effects analysis (FMEA)

Example: A system has three components in series. If each has reliability 0.95, what's the system reliability?

$$P(System) = P(C1) × P(C2) × P(C3) = 0.95 × 0.95 × 0.95 = 0.857$$

7. Weather Forecasting and Environmental Science

Weather Prediction:

• Probability of Precipitation: Based on atmospheric conditions
• Hurricane Tracking: Probability of landfall and intensity changes
• Climate Modeling: Long-term probability-based predictions

Environmental Risk Assessment:

• Flood Risk: Probability based on historical data and geography
• Earthquake Hazard: Seismic probability calculations
• Air Quality: Probability of exceeding pollution thresholds

8. Artificial Intelligence and Machine Learning

Machine Learning Algorithms:

• Classification: Probability-based decision making
• Natural Language Processing: Probability of word sequences
• Computer Vision: Probability of object recognition

Bayesian Networks:

Probability in Daily Life

1. Decision Making

Expected Utility Theory:

$$E[U] = \sum P_i × U_i$$

Where $P_i$ is probability of outcome $i$ and $U_i$ is utility of outcome $i$.

Example: Should you buy a lottery ticket for RM1 with 1 in 10 million chance to win RM5 million?

E[U] = (1/10,000,000) × 5,000,000 + (9,999,999/10,000,000) × (-1) = 0.5 + (-0.9999999) ≈ -RM0.50

2. Risk Management

Insurance Decisions:

• Calculate expected loss vs. premium cost
• Probability of catastrophic events
• Risk tolerance assessment

Investment Decisions:

• Expected returns based on probability distributions
• Risk-return trade-off analysis
• Portfolio diversification for risk reduction

3. Games and Gambling

Understanding House Edge:

• Roulette: House edge ≈ 5.26% (American)
• Blackjack: House edge ≈ 0.5% (with basic strategy)
• Slot machines: House edge typically 2-15%

Expected Value Calculation: For a RM10 bet with 50% chance to win RM20: E[X] = (0.5 × 20) + (0.5 × -10) = 10 - 5 = RM5 Expected profit per bet = RM5 - RM10 = -RM5

Important Terms

Summary Points

• Independent Events: P(A and B) = P(A) × P(B)
• Dependent Events: P(A and B) = P(A) × P(B|A)
• Mutually Exclusive: P(A or B) = P(A) + P(B)
• Non-Mutually Exclusive: P(A or B) = P(A) + P(B) - P(A and B)
• Tree Diagrams: Useful for visualizing complex probabilities
• Sample Space: Total possible outcomes
• Key Principle: Multiply along branches, add across branches
• Bayes' Theorem: P(A|B) = P(B|A) × P(A) / P(B)
• Expected Value: E[X] = Σ P(x) × x
• Complement Rule: P(A') = 1 - P(A)
• Probability Range: 0 ≤ P(A) ≤ 1

Practice Tips for SPM Students

1. Identify Event Types

• Learn to distinguish between independent and dependent events
• Recognize mutually exclusive vs. non-mutually exclusive events
• Practice identifying the correct probability rule for each scenario

2. Master Tree Diagrams

• Practice drawing accurate tree diagrams
• Learn to calculate probabilities along branches
• Understand how to use tree diagrams for conditional probability

3. Calculate Probabilities

• Practice multiplication and addition rule applications
• Learn to handle fractions in probability calculations
• Understand probability notation (P(A), P(B|A), etc.)

4. Common Mistakes to Avoid

• Confusing independent and dependent events
• Forgetting to subtract intersection in addition rule
• Misidentifying mutually exclusive events
• Incorrect conditional probability calculations

SPM Exam Tips

Paper 1 (Multiple Choice)

• Look for key words indicating event relationships
• Practice quick probability calculations
• Remember the fundamental probability rules
• Use elimination method for complex questions

Paper 2 (Structured)

• Show all probability calculations clearly
• Draw tree diagrams when appropriate
• Explain your reasoning for event classification
• Use proper probability notation in your answers

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Did You Know? Probability theory was developed in the 17th century by mathematicians like Blaise Pascal and Pierre de Fermat as they analyzed games of chance. Their work laid the foundation for modern probability theory and statistics!

Fun Fact: The probability of being dealt a royal flush in poker is approximately 1 in 649,740 - making it one of the rarest hands in the card game!

Next Chapter: In Chapter 10, you'll explore consumer mathematics focusing on financial management, including budgeting, saving, and investment strategies.

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Probability isn't just about numbers - it's about understanding uncertainty and making informed decisions in an uncertain world. From weather forecasts to medical diagnoses, from game strategies to financial markets, probability concepts help us navigate the randomness of life with confidence and wisdom.

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• Chapter 8: Introduction to Statistics
• Chapter 10: Financial Management