Unit: Data Handling & Analysis
Chapter: Chance & Probability
Reference: – Compound Events, Independent and Dependent Events, Mutually Exclusive Events, Addition Rule of Probability, Multiplication Rule of Probability, Probability of "Or" (Union), Probability of "And" (Intersection), Tree Diagrams, Sample Space for Multiple Events, Probability Without Replacement, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- Compound Events and How to Calculate Their Probability
- Difference Between Independent and Dependent Events
- The Addition Rule (Probability of A or B)
- The Multiplication Rule (Probability of A and B)
- Using Tree Diagrams to Find Probabilities
- Probability Without Replacement
Introduction to Chance & Probability – 2
Definition
In Probability – 1, we studied simple events (like rolling one die or flipping one coin). In Probability – 2, we study compound events – events that involve two or more simple events combined. We learn how to find the probability that event A AND event B both happen, or that event A OR event B happens.
When we study compound probability, we essentially ask:
"What is the chance that both events happen? What is the chance that at least one of them happens?"
To answer these questions, we need to understand whether events are independent or dependent, mutually exclusive or not, and use the correct rules.
Importance of Compound Probability
- Used in weather forecasting (chance of rain AND wind)
- Used in games (probability of drawing two aces in a row)
- Essential for risk analysis (probability of multiple failures)
- Used in genetics (probability of inheriting multiple traits)
- Foundation for advanced statistics and data science
Example
If you flip a coin and roll a die, what is the probability of getting heads AND rolling a 6?
The two events are independent. P(heads) = 1/2, P(6) = 1/6, so P(heads and 6) = 1/2 × 1/6 = 1/12.
Subtopics
1. Simple Events vs Compound Events
Simple Event: An event that cannot be broken down into smaller events. Example: rolling a 4 on a die.
Compound Event: An event that consists of two or more simple events. Example: rolling a 4 on a die AND flipping heads on a coin.
Compound events can be combined using "AND" (both happen) or "OR" (at least one happens).
2. Independent Events vs Dependent Events
Independent Events: The outcome of one event does NOT affect the outcome of the other event.
Examples of Independent Events:
- Flipping a coin and rolling a die
- Drawing a card from a deck and then replacing it before drawing again
- The weather in two different cities on the same day
For independent events: P(A and B) = P(A) × P(B)
Dependent Events: The outcome of one event DOES affect the outcome of the other event.
Examples of Dependent Events:
- Drawing two cards from a deck without replacement
- Picking two marbles from a bag without putting the first back
- The probability of being late to school depends on whether the bus is on time
For dependent events: P(A and B) = P(A) × P(B given that A has occurred)
P(B given A) is called conditional probability and is written as P(B|A).
3. Mutually Exclusive Events
Mutually Exclusive Events: Two events that cannot happen at the same time. They have no outcomes in common.
Examples of Mutually Exclusive Events:
- Rolling a 3 and rolling a 5 on the same die roll (cannot happen together)
- Flipping heads and flipping tails on the same coin flip
- Drawing a heart and drawing a spade in one card draw
For mutually exclusive events: P(A or B) = P(A) + P(B)
Non-Mutually Exclusive Events: Events that can happen at the same time. They share some outcomes.
Examples of Non-Mutually Exclusive Events:
- Drawing a heart and drawing a king (the king of hearts is both)
- Rolling an even number and rolling a number greater than 3 (the number 4 and 6 are both)
For non-mutually exclusive events: P(A or B) = P(A) + P(B) – P(A and B)
4. The Addition Rule (Probability of A or B)
The addition rule tells us the probability that event A OR event B occurs (or both).
Case 1 – Mutually Exclusive Events:
P(A or B) = P(A) + P(B)
Example: What is the probability of rolling a 2 or a 5 on a die?
P(2) = 1/6, P(5) = 1/6, and they cannot both happen → P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3
Case 2 – Non-Mutually Exclusive Events:
P(A or B) = P(A) + P(B) – P(A and B)
Example: What is the probability of drawing a heart or a king from a standard deck?
P(heart) = 13/52, P(king) = 4/52, P(heart and king) = 1/52 (the king of hearts)
P(heart or king) = 13/52 + 4/52 – 1/52 = 16/52 = 4/13
5. The Multiplication Rule (Probability of A and B)
The multiplication rule tells us the probability that event A AND event B both occur.
Case 1 – Independent Events:
P(A and B) = P(A) × P(B)
Example: What is the probability of flipping tails and rolling a 3?
P(tails) = 1/2, P(3) = 1/6 → P(tails and 3) = 1/2 × 1/6 = 1/12
Case 2 – Dependent Events:
P(A and B) = P(A) × P(B given A)
Example: A bag has 3 red and 5 blue marbles. You draw two marbles without replacement. What is the probability both are red?
P(first red) = 3/8
After removing one red, 2 red and 5 blue remain (7 marbles total).
P(second red | first red) = 2/7
P(both red) = 3/8 × 2/7 = 6/56 = 3/28
6. Probability Without Replacement (A Special Case of Dependent Events)
Without replacement means the first item is not returned to the set before drawing the second item. This makes events dependent because the total number of items decreases and the composition changes.
Important: After each draw without replacement, both the total count and the count of favorable outcomes decrease.
Example – Marbles without replacement: A bag has 4 red, 3 blue, and 5 green marbles. Two marbles are drawn without replacement. Find P(both blue).
Total marbles = 12
P(first blue) = 3/12 = 1/4
After removing one blue: 2 blue left, total 11 marbles
P(second blue | first blue) = 2/11
P(both blue) = 1/4 × 2/11 = 2/44 = 1/22
Example – Drawing cards without replacement: What is the probability of drawing two aces from a standard deck without replacement?
P(first ace) = 4/52 = 1/13
After removing one ace: 3 aces left, total 51 cards
P(second ace | first ace) = 3/51 = 1/17
P(both aces) = 1/13 × 1/17 = 1/221
7. Tree Diagrams for Probability
A tree diagram is a visual tool that shows all possible outcomes of compound events and their probabilities. Each branch represents a possible outcome, and the probabilities along each path are multiplied to find the probability of that path.
Steps to Use a Tree Diagram:
Step 1: Draw branches for the first event, label each outcome with its probability.
Step 2: From each branch, draw branches for the second event, label with conditional probabilities.
Step 3: Multiply probabilities along each path to find the probability of that sequence.
Step 4: Add probabilities of paths that match the event you are interested in.
Example – Tree Diagram for Two Coin Flips:
First flip: H (1/2) and T (1/2)
From H: second flip H (1/2) and T (1/2)
From T: second flip H (1/2) and T (1/2)
Paths: HH (1/4), HT (1/4), TH (1/4), TT (1/4)
Example – Tree Diagram for Marbles Without Replacement:
Bag: 3 red, 2 blue. Draw two marbles without replacement.
First draw: R (3/5) and B (2/5)
From R (3/5): second draw – R (2/4 = 1/2), B (2/4 = 1/2)
From B (2/5): second draw – R (3/4), B (1/4)
Multiply along paths:
R then R: 3/5 × 1/2 = 3/10
R then B: 3/5 × 1/2 = 3/10
B then R: 2/5 × 3/4 = 6/20 = 3/10
B then B: 2/5 × 1/4 = 2/20 = 1/10
Check sum: 3/10 + 3/10 + 3/10 + 1/10 = 10/10 = 1 ✓
Solved Examples
Example 1 – Independent Events: A coin is flipped and a die is rolled. What is the probability of getting tails and an even number?
Solution: P(tails) = 1/2, P(even) = 3/6 = 1/2
Events are independent. P(tails and even) = 1/2 × 1/2 = 1/4
Answer: 1/4
Example 2 – Dependent Events (Without Replacement): A box contains 5 red pens and 7 blue pens. Two pens are drawn without replacement. What is the probability both are blue?
Solution: Total pens = 12
P(first blue) = 7/12
After removing one blue: 6 blue left, total 11 pens
P(second blue | first blue) = 6/11
P(both blue) = 7/12 × 6/11 = 42/132 = 7/22
Answer: 7/22
Example 3 – Mutually Exclusive Events: A die is rolled once. What is the probability of rolling a 2 or a 5?
Solution: P(2) = 1/6, P(5) = 1/6. Cannot roll both at once (mutually exclusive).
P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3
Answer: 1/3
Example 4 – Non-Mutually Exclusive Events: A card is drawn from a standard deck. What is the probability it is a face card or a heart?
Solution: P(face card) = 12/52 (Jack, Queen, King of each suit – 3 per suit × 4 suits = 12)
P(heart) = 13/52
P(face card and heart) = 3/52 (Jack, Queen, King of hearts)
P(face card or heart) = 12/52 + 13/52 – 3/52 = 22/52 = 11/26
Answer: 11/26
Example 5 – Using Complement: Two dice are rolled. What is the probability of getting at least one 6?
Solution: It is easier to find P(no six) and subtract from 1.
P(no six on one die) = 5/6
For two dice, P(no six on either) = 5/6 × 5/6 = 25/36
P(at least one six) = 1 – 25/36 = 11/36
Answer: 11/36
Example 6 – Tree Diagram Problem: A bag has 4 red and 6 yellow marbles. Two marbles are drawn without replacement. Use a tree diagram to find P(one of each color).
Solution:
First draw: R (4/10 = 2/5), Y (6/10 = 3/5)
From R (2/5): second draw – R (3/9 = 1/3), Y (6/9 = 2/3)
From Y (3/5): second draw – R (4/9), Y (5/9)
Paths that give one of each color: R then Y, and Y then R
R then Y: 2/5 × 2/3 = 4/15
Y then R: 3/5 × 4/9 = 12/45 = 4/15
P(one of each) = 4/15 + 4/15 = 8/15
Answer: 8/15
Common Mistakes to Avoid
Mistake 1 – Treating dependent events as independent
When drawing without replacement, the probabilities change.
Correct understanding: After each draw without replacement, the total decreases and the composition changes.
Mistake 2 – Using addition rule when events are not mutually exclusive
P(A or B) = P(A) + P(B) only works if A and B cannot happen together.
Correct understanding: If events can happen together, subtract P(A and B).
Mistake 3 – Forgetting to subtract the overlap in non-mutually exclusive events
P(A or B) = P(A) + P(B) – P(A and B). The overlap is counted twice if we just add.
Correct understanding: Always check if events share outcomes.
Mistake 4 – Confusing "or" with "and"
"And" means both happen (multiply probabilities carefully). "Or" means at least one happens (add with subtraction).
Correct understanding: Read the problem carefully to know which rule applies.
Mistake 5 – Not simplifying conditional probability correctly
P(B|A) = P(B given A occurred). After event A, the sample space changes.
Correct understanding: Recalculate total outcomes and favorable outcomes based on the new information.
Mistake 6 – Multiplying probabilities that are not independent
For dependent events, you cannot simply multiply P(A) and P(B). You need P(A) × P(B|A).
Correct understanding: Identify dependence first, then use the correct formula.
Mistake 7 – Forgetting that "without replacement" changes the denominator
When drawing two marbles without replacement, the second draw's total is one less.
Correct understanding: Adjust both numerator and denominator for each subsequent draw.
Quick Reference Summary
Independent Events: P(A and B) = P(A) × P(B)
Dependent Events: P(A and B) = P(A) × P(B|A)
Mutually Exclusive Events: P(A or B) = P(A) + P(B)
Non-Mutually Exclusive Events: P(A or B) = P(A) + P(B) – P(A and B)
Without Replacement: Events become dependent; update counts after each draw
Tree Diagrams: Multiply along branches, add across branches that match the event
Complement Rule (for "at least one"): P(at least one) = 1 – P(none)
Key Words:
"And" → Multiply (check independence/dependence)
"Or" → Add (check mutually exclusive or not)
"Without replacement" → Dependent events
"With replacement" → Independent events