Probability Scales

Welcome to the World of Probability!

Have you ever wondered what the "chance" of rain is, or how likely you are to win a game of Rock, Paper, Scissors? That is exactly what Probability is all about! In this chapter, we are going to learn how to measure how likely something is to happen using a Probability Scale.

Don't worry if you find numbers a bit tricky—we will start with words first and then move on to simple numbers and fractions. By the end of these notes, you’ll be a pro at predicting the future (mathematically speaking)!

1. What is Probability?

Probability is a measure of how likely it is that an event (something happening) will occur. We use a scale to show this. Think of it like a volume slider on your phone: on one end, there is no sound at all (it won't happen), and on the other end, it is at full volume (it definitely will happen).

The Word Scale

Before we use numbers, we can use words to describe the chance of something happening. Here are the five key words you need to know:

• Impossible: The event definitely will not happen. (Example: You seeing a live dinosaur walking down your street today.)
• Unlikely: There is a small chance, but it probably won't happen. (Example: Winning a big prize in a raffle with 1,000 tickets when you only have 1 ticket.)
• Even Chance: It is just as likely to happen as it is not to happen. We often call this a "50/50" chance. (Example: Getting "Heads" when you flip a coin.)
• Likely: There is a good chance it will happen, but it’s not guaranteed. (Example: It raining in the UK during December.)
• Certain: The event definitely will happen. (Example: The sun rising tomorrow.)

Key Takeaway: Probability always moves from Impossible on the left to Certain on the right.

2. The Number Scale (0 to 1)

In Mathematics, we give these words actual values. The most important rule in probability is this: Probability is always between 0 and 1.

• If the probability is 0, it is Impossible.
• If the probability is 1, it is Certain.
• If the probability is 0.5, it is an Even Chance.

Memory Aid: Think of the "1" as representing 100%. If you are 100% sure, the probability is 1. If there is 0% chance, the probability is 0.

Different Ways to Write Probabilities

You can write a probability as a fraction, a decimal, or a percentage. They all mean the same thing!

• Even Chance = \( \frac{1}{2} \) or \( 0.5 \) or \( 50\% \)
• Unlikely = Anything between \( 0 \) and \( 0.5 \) (like \( 0.2 \) or \( \frac{1}{4} \))
• Likely = Anything between \( 0.5 \) and \( 1 \) (like \( 0.8 \) or \( 75\% \))

Quick Review:
Is a probability of \( 1.2 \) possible? No! Probability can never be bigger than 1.
Is a probability of \( -0.5 \) possible? No! Probability can never be less than 0.

3. How to Calculate Simple Probability

Sometimes, we need to find the exact spot on the scale for an event. We can do this with a simple formula:

Probability of an event = \( \frac{\text{Number of ways it can happen}}{\text{Total number of possible outcomes}} \)

Example: Rolling a Dice

If you roll a standard six-sided dice, what is the probability of rolling a \( 4 \)?

1. How many ways can it happen? There is only one "\( 4 \)" on the dice. (So, \( 1 \))
2. What are the total outcomes? There are \( 6 \) sides in total. (So, \( 6 \))
3. The Probability: \( P(4) = \frac{1}{6} \)

Where would this go on our scale? Since \( \frac{1}{6} \) is much less than half (\( \frac{3}{6} \)), we would say it is Unlikely.

Did you know? The word "Probability" comes from the Latin word probabilis, which means "provable" or "credible"!

4. Placing Events on a Probability Scale

When you are asked to "mark" an event on a scale, follow these steps:

Step 1: Look at the total number of options (this is the end of your fraction).
Step 2: Look at how many options match what you are looking for (this is the top of your fraction).
Step 3: Convert it to a decimal if that helps you visualize it.
Step 4: Draw an arrow on the line between 0 and 1.

Example: A Bag of Marbles

A bag has \( 10 \) marbles: \( 7 \) are Red and \( 3 \) are Blue.
The probability of picking a Red marble is \( \frac{7}{10} \) or \( 0.7 \).
On a scale, this would be marked slightly to the right of the middle (\( 0.5 \)), in the Likely section.

5. Common Mistakes to Avoid

• Don't mix up the ends: Remember, \( 0 \) is nothing (Impossible), \( 1 \) is everything (Certain).
• The "50/50" Trap: Students often think everything is an "Even Chance" (either it happens or it doesn't). But this is only true if both outcomes are equally likely, like a fair coin. A lottery has two outcomes (win or lose), but they are definitely not equally likely!
• Using "Very Likely": In some exams, stick to the main five words (Impossible, Unlikely, Even, Likely, Certain) unless the question asks for more detail.

Key Takeaway: Always check your answer. If your probability is a fraction where the top number is bigger than the bottom number (like \( \frac{5}{4} \)), something has gone wrong!

Summary: The Quick Guide

Impossible = \( 0 \)
Unlikely = Between \( 0 \) and \( 0.5 \)
Even Chance = \( 0.5 \) or \( \frac{1}{2} \)
Likely = Between \( 0.5 \) and \( 1 \)
Certain = \( 1 \)

Final Tip: If you're feeling stuck, draw the scale line! Seeing the line from 0 to 1 makes it much easier to decide where an event belongs. You've got this!