Welcome to the World of Probability!

Ever wondered what the chances are of winning a game, flipping a coin and getting heads, or what the weather forecast means when it says "80% chance of rain"? That's all about probability! Probability is the branch of mathematics that tells us how likely something is to happen. It's a super useful skill that helps us make smarter decisions in games, in science, and in everyday life. Don't worry if this sounds complicated, we're going to break it down into easy, fun steps. Let's get started!


1. Certain, Impossible, and Random Events

First, let's learn about the three main types of events, or "things that can happen", in probability.

Certain Events

A certain event is something that is guaranteed to happen. There is no doubt about it!

Examples:

  • If you drop a book, it will fall down (thanks, gravity!).
  • The day after Monday is always Tuesday.
  • If a bag only contains red balls, you are certain to pick a red ball.
Impossible Events

An impossible event is something that can never, ever happen. There's zero chance!

Examples:

  • A fish will start singing opera.
  • You can roll a '7' on a normal six-sided die.
  • If a bag only contains red balls, it's impossible to pick a blue ball.
Random Events

A random event is any event where the outcome is not known for sure before it happens. It might happen, or it might not. This is where probability gets really interesting!

Examples:

  • Flipping a coin and getting 'Tails'.
  • Winning a lucky draw.
  • Your favourite football team winning their next match.
Key Takeaway

Events can be certain (will definitely happen), impossible (will never happen), or random (might happen). Most of the things we study in probability are random events.


2. What is Probability? (The Chance-o-Meter!)

Think of probability as a "Chance-o-Meter" that measures how likely an event is. This meter goes from 0 to 1.

  • A probability of 0 means the event is impossible.
  • A probability of 1 means the event is certain.
  • A probability of 0.5 (or 1/2 or 50%) means the event has an equal chance of happening or not happening, like getting 'Heads' when you flip a coin.

We can write probability as a fraction, a decimal, or a percentage.

The Magic Formula

To calculate the probability of a random event, we use one simple formula. It's the most important tool in your probability toolbox!

$$P(\text{event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$$

Let's break that down:

  • Favourable Outcomes: This is a fancy way of saying "the outcomes you are interested in" or "the ways you can win".
  • Total Possible Outcomes: This is every single possible thing that could happen.
Example: Rolling a Die

Imagine you want to find the probability of rolling a 4 on a standard six-sided die.
- The number of favourable outcomes is 1 (because there's only one face with a '4').
- The total number of possible outcomes is 6 (because the die can land on 1, 2, 3, 4, 5, or 6).

$$P(\text{rolling a 4}) = \frac{1}{6}$$

So, the probability is 1 out of 6. Easy, right?


Quick Review Box

Probability is a measure of chance between 0 and 1.
0 = Impossible
1 = Certain
Formula: P(event) = Favourable Outcomes / Total Outcomes


3. Listing All the Outcomes: The Sample Space

Before we can use our formula, we need to know all the possible outcomes. The complete list of all possible outcomes of an experiment is called the sample space. Getting this right is the most important first step!

For example, the sample space for rolling a die is {1, 2, 3, 4, 5, 6}.

When things get more complex, like flipping two coins or rolling two dice, we need organised ways to find the sample space. Let's learn two great methods!

Method 1: Using a Table (or Grid)

Tables are perfect when you have two separate things happening at the same time (like rolling two dice).

Example: Rolling Two Dice

Let's find the sample space for the sum of the numbers when rolling two dice.

Step 1: Set up a table. One die's outcomes go along the top, and the other's go down the side.
Step 2: Fill in the middle by adding the numbers from the corresponding row and column.

Die 2

      + | 1 | 2 | 3 | 4 | 5 | 6
    ---|---|---|---|---|---|---
    1 | 2 | 3 | 4 | 5 | 6 | 7
D 2 | 3 | 4 | 5 | 6 | 7 | 8
i 3 | 4 | 5 | 6 | 7 | 8 | 9
e 4 | 5 | 6 | 7 | 8 | 9 | 10
    5 | 6 | 7 | 8 | 9 | 10| 11
1 6 | 7 | 8 | 9 | 10| 11| 12

The total number of possible outcomes is 6 × 6 = 36. Now we can easily answer questions like, "What's the probability of the sum being 7?"
Just count the 7s in the table! There are 6 of them. So, the probability is:

$$P(\text{sum is 7}) = \frac{6}{36} = \frac{1}{6}$$

Method 2: Using a Tree Diagram

Tree diagrams are fantastic for experiments that happen in stages or steps (like flipping a coin three times).

Example: Flipping a Coin Two Times

Let's find the sample space for flipping a coin twice.

Step 1: Draw the first "branches" for the first flip (Heads or Tails).
Step 2: From the end of each of those branches, draw the branches for the second flip.
Step 3: List the final outcomes by following each path from start to finish.

1st Flip       2nd Flip       Outcomes
      ,- - - Heads     ->   (Heads, Heads) or HH
  Heads
      `- - - Tails       ->   (Heads, Tails) or HT

      ,- - - Heads     ->   (Tails, Heads) or TH
  Tails
      `- - - Tails       ->   (Tails, Tails) or TT

The sample space is {HH, HT, TH, TT}. The total number of possible outcomes is 4. Now we can answer, "What is the probability of getting at least one Head?"
Three outcomes have at least one Head (HH, HT, TH). So the probability is:

$$P(\text{at least one Head}) = \frac{3}{4}$$

Key Takeaway

The sample space is the list of all possible outcomes. Use a table for two simultaneous events and a tree diagram for events that happen in stages. Don't worry if this seems tricky at first, practice is all you need!


4. Let's Solve Some Problems!

Time to use our new skills. We will always follow these steps:
Step 1: Figure out the total number of possible outcomes (the sample space).
Step 2: Count the number of favourable outcomes.
Step 3: Put them into the probability formula and simplify!

Problem 1: Picking a Marble

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of picking a blue marble?

Step 1: Total Outcomes. Total number of marbles is 5 + 3 + 2 = 10.
Step 2: Favourable Outcomes. The number of blue marbles is 3.
Step 3: Calculate.

$$P(\text{blue marble}) = \frac{3}{10}$$

Problem 2: Using our Dice Table

You roll two dice. What is the probability that the sum is greater than 9?

Step 1: Total Outcomes. From our table, we know there are 36 total outcomes.
Step 2: Favourable Outcomes. Let's look at the table and count the sums greater than 9 (so, 10, 11, or 12). We find: (4,6), (5,5), (6,4), (5,6), (6,5), (6,6). That's 6 favourable outcomes.
Step 3: Calculate.

$$P(\text{sum > 9}) = \frac{6}{36} = \frac{1}{6}$$

Common Mistakes to Avoid
  • Forgetting to simplify the fraction. Always give your answer in the simplest form (e.g., 6/36 becomes 1/6).
  • Incorrectly counting the total outcomes. Be careful and systematic. Using a table or tree diagram helps prevent mistakes!

5. Taking it a Step Further: What to Expect (Expectation)

(This is a slightly more advanced topic, but it's very cool!)

The expectation (or expected value) is what you would expect to be the average result if you ran an experiment many, many times. It helps us figure out if a game is fair or what the long-term outcome might be.

Imagine a simple game: You roll a die. If you roll a 6, you win $12. If you roll any other number, you win nothing. Is it worth paying $1 to play this game?

How to Calculate Expectation

Step 1: List each possible outcome (value, like $12 or $0).
Step 2: Find the probability of each outcome.
Step 3: Multiply each outcome's value by its probability.
Step 4: Add those results together.

Let's solve our game problem:

Outcome 1: Win $12. The probability of this is P(rolling a 6) = 1/6.
Outcome 2: Win $0. The probability of this is P(not rolling a 6) = 5/6.

Step 3:
For outcome 1: $$12 \times \frac{1}{6} = $2$$ For outcome 2: $$0 \times \frac{5}{6} = $0$$

Step 4:
Expectation = $2 + $0 = $2.

This means that if you played this game over and over, you would expect to win an average of $2 per game. Since it only costs $1 to play, it's a very good game to play!

Did you know?

The concept of expectation is used by insurance companies to set prices and by businesses to make decisions about risk. It's powerful stuff!


Chapter Summary

Congratulations! You've learned the fundamentals of probability.

  • Events can be certain, impossible, or random.
  • Probability is measured on a scale from 0 (impossible) to 1 (certain).
  • The key formula is P(event) = Favourable / Total.
  • A sample space lists all possible outcomes. We can find it using a table or a tree diagram.
  • Expectation tells us the long-term average outcome of an experiment.

Keep practicing, and you'll become a probability pro in no time!