Welcome to the World of Probability!
Hello future mathematician! This chapter is all about Probability—the mathematics of chance. Don't worry if words like 'statistics' sound scary; probability is deeply rooted in everyday life, from checking the weather forecast to planning your schedule.
We will learn how to calculate, visualize, and understand the likelihood of things happening. These notes are designed to break down every concept into simple steps. Let's dive in!
1. The Foundation: Defining and Measuring Probability
What is Probability?
Probability measures how likely an event is to occur. We always express it as a number between 0 and 1 (or 0% and 100%).
- 0 means the event is Impossible (it will never happen). Example: Rolling a standard die and getting a 7.
- 1 (or 100%) means the event is Certain (it will definitely happen). Example: The sun rising tomorrow.
- 0.5 (or 50%) means the event is Even Chance (just as likely to happen as not).
Calculating Theoretical Probability
When we assume all outcomes are equally likely (like a fair coin or a fair die), we use Theoretical Probability.
The fundamental formula is:
\(P(A) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\)
Example: What is the probability of rolling an even number on a standard 6-sided die?
- Favourable outcomes (2, 4, 6): 3
- Total possible outcomes (1, 2, 3, 4, 5, 6): 6
- \(P(\text{Even}) = \frac{3}{6} = \frac{1}{2}\) or 0.5
Experimental Probability (Relative Frequency)
Sometimes we can't calculate theoretical probability easily (like predicting the outcome of a complex sporting event), so we rely on real-world data, which gives us Experimental Probability. This is also known as Relative Frequency.
Relative Frequency (RF) is based on trials or experiments:
\(\text{RF} = \frac{\text{Number of successful trials}}{\text{Total number of trials performed}}\)
Did you know? As you perform more and more trials, the experimental probability usually gets closer and closer to the theoretical probability. This is a very important concept called the Law of Large Numbers!
Quick Review: Key Takeaway
Probability is a measurement (0 to 1). Theoretical uses the formula based on possibilities; Experimental uses observed data (Relative Frequency).
2. The Basic Rules of Probability
Rule 1: The Sum of All Probabilities
If you list every single possible outcome in an experiment (this list is called the Sample Space), the probabilities of all those outcomes must add up to 1.
\(\sum P = 1\)
Rule 2: The Complement Rule (The "Not" Rule)
The probability of an event happening plus the probability of that event not happening must equal 1.
We use the notation \(A'\) (A Prime) or \(A^c\) (A Complement) to mean 'Not A'.
\(P(A') = 1 - P(A)\)
Analogy: If there is a 0.25 chance of your bus being late (\(P(L)\)), then the chance of it being on time (\(P(L')\)) is \(1 - 0.25 = 0.75\).
3. Visualizing Outcomes: Sample Space Diagrams
When an experiment involves two separate actions (like rolling two dice, or flipping a coin and rolling a die), a Sample Space Diagram (usually a table) is the clearest way to list all possible outcomes.
Step-by-Step: Creating a Sample Space Table
- List the outcomes of the first action along the top (x-axis).
- List the outcomes of the second action along the side (y-axis).
- Fill in the table with the combined results (often the sum or product of the two results).
Example: Rolling two standard dice and adding the scores.
The table would show 36 total outcomes (6 x 6).
To find \(P(\text{Total score of 7})\):
You would count the boxes that equal 7 (there are 6 of them: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)).
\(P(\text{Score of 7}) = \frac{6}{36} = \frac{1}{6}\)
Avoid This Common Mistake!
When rolling two dice, students sometimes think \(P(\text{total 7})\) is the same as \(P(\text{total 2})\). It is NOT! A sample space diagram shows that 7 is much more likely because there are 6 ways to get it, but only 1 way to get a 2 (1 + 1).
4. Types of Events and Key Rules
4.1. Mutually Exclusive Events (The Addition Rule)
Mutually Exclusive events are things that cannot happen at the same time. They have no overlap.
- Example: Flipping a coin once and getting a 'Head' and getting a 'Tail' are mutually exclusive.
If two events A and B are mutually exclusive, the probability that A OR B occurs is the sum of their individual probabilities.
P(A or B) = P(A) + P(B)
Example: A bag contains 5 red, 3 blue, and 2 green marbles (total 10). What is the probability of picking a Red OR a Green marble?
\(P(\text{Red}) = 5/10\)
\(P(\text{Green}) = 2/10\)
\(P(\text{Red or Green}) = 5/10 + 2/10 = 7/10\)
4.2. Independent Events (The Multiplication Rule)
Independent Events are events where the outcome of the first event does not affect the outcome of the second event.
- Example: Flipping a coin and then rolling a die. The coin flip doesn't change the chances of the die roll.
If two events A and B are independent, the probability that A AND B occurs is the product of their individual probabilities.
P(A and B) = P(A) \(\times\) P(B)
Example: What is the probability of rolling a 6 on a die AND flipping a Head on a coin?
\(P(6) = 1/6\)
\(P(H) = 1/2\)
\(P(6 \text{ and } H) = (1/6) \times (1/2) = 1/12\)
5. Using Tree Diagrams for Sequential Events
Tree Diagrams are fantastic visual tools used for experiments that happen in sequence (one after the other).
How to Build and Use a Tree Diagram
- Start with Branches: Draw branches for the first event, labeling each branch with the outcome and its probability.
- Secondary Branches: From the end of each first branch, draw new branches for the second event, labeling their probabilities.
- Find the Final Probabilities: To find the probability of a sequence of events (e.g., Event A then Event B), you multiply the probabilities along the path (or branch) you take.
- Calculate Total Probabilities: If you are looking for multiple successful outcomes (e.g., both Head-Tail and Tail-Head), you add the probabilities of the relevant end points.
Scenario: With and Without Replacement
This is the most common place where students slip up! Take a deep breath—the key is to check if the sample space changes.
5.1. With Replacement (Independent Events)
If you pick an item and then put it back, the total number of items and the probability for the second event remain the same. The events are independent.
Example: Picking a marble, noting its color, and putting it back before picking a second one.
5.2. Without Replacement (Dependent Events)
If you pick an item and do NOT put it back, the total number of items decreases by 1, and the probabilities for the second event will change. The events are now dependent.
Example: A bag has 4 Red and 6 Blue marbles (Total 10). You pick two without replacement.
- \(P(\text{1st is Red}) = 4/10\)
- If the 1st was Red, the bag now has 3 Red and 6 Blue (Total 9).
- \(P(\text{2nd is Red}) = 3/9\)
- Therefore, \(P(\text{Red and Red}) = (4/10) \times (3/9) = 12/90\)
Memory Aid for Tree Diagrams
Multiply along the branches (AND).
Add at the end points (OR).
6. Visualizing Probability with Venn Diagrams
Venn Diagrams are used to visualize the relationships between different sets or events. They are especially helpful for non-mutually exclusive events (events that can overlap).
- The large rectangle represents the Sample Space (\(\mathcal{E}\) or S)—the total population or total number of outcomes.
- The circles represent specific events (A, B, etc.).
Key Set Notation for Probability
6.1. Intersection (\(A \cap B\)) - AND
This is the central overlapping area where A and B both occur.
\(P(A \cap B)\) means the probability of A AND B happening.
6.2. Union (\(A \cup B\)) - OR
This includes everything in A, everything in B, and the overlap.
\(P(A \cup B)\) means the probability of A OR B happening (or both).
6.3. Complement (\(A'\)) - NOT
This is the area outside the circle A (but still inside the rectangle \(\mathcal{E}\)).
The General Addition Formula
If events A and B are not mutually exclusive (they overlap), we cannot just add \(P(A) + P(B)\), because we would count the intersection (\(A \cap B\)) twice!
We must use the general formula:
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Example: If 10 people like Apples (A) and 8 like Bananas (B), and 3 people like both (\(A \cap B\)), we find the total unique people who like A or B by calculating \(10 + 8 - 3 = 15\).
Venn Diagram Steps
- Always start by filling in the intersection (\(A \cap B\)) first.
- Calculate the unique parts: For A only, subtract the overlap from the total A: \(P(A \text{ only}) = P(A) - P(A \cap B)\).
- Calculate the area outside the circles (The Complement of the Union): This represents those who are neither A nor B.
7. Final Review and Encouragement
Quick Probability Checklist
- Is the answer between 0 and 1? (If not, re-check your calculations!)
- Are the events mutually exclusive? If yes, Add (OR).
- Are the events independent? If yes, Multiply (AND).
- If using a tree diagram, did you remember to update the total outcomes for 'Without Replacement' questions?
- If using a Venn diagram, did you subtract the overlap when calculating the union?
You have mastered the fundamental concepts of IGCSE Probability! Remember, probability is about logical counting. Keep practicing those tree diagrams and sample spaces, and you will find these questions straightforward. You've got this!