The Probability Scale
Place events on the 0-to-1 probability scale, convert freely between fractions, decimals and percentages, and compare two probabilities written in different forms.
Printable Worksheets
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Think First
Put these events in order from least to most likely: rolling a 7 on a standard die, flipping heads on a fair coin, drawing an Ace from a full deck, it raining somewhere in Australia today. Can you place each one on a scale from 0 to 1?
The Probability Scale
Every probability fits somewhere between 0 (absolutely impossible) and 1 (completely certain). Knowing where events sit on this scale lets us compare likelihoods with precision, not just "pretty likely" but exactly 0.67.
What You'll Master
- Describe events as impossible, unlikely, even chance, likely, or certain
- Place events accurately on a probability number line from 0 to 1
- Convert probability values between fractions, decimals, and percentages
- Compare the likelihood of events using their probability values
- Order a list of events from least to most likely
Words You Need
⚠ Spot the Trap
"Unlikely" does NOT mean "impossible". An event with probability 0.01 is very unlikely, but it can still happen. Similarly, "likely" does NOT mean "certain". $P = 0.99$ means the event will almost always happen, but there is still a 1% chance it won't. Only $P = 0$ is impossible and only $P = 1$ is certain. Every value in between represents genuine uncertainty.
The Five Points on the Scale
| Label | P value | Example |
|---|---|---|
| Impossible | 0 | Rolling a 7 on a standard die |
| Unlikely | 0 < P < 0.5 | Drawing an Ace from a deck (≈ 0.077) |
| Even Chance | 0.5 | Flipping heads on a fair coin |
| Likely | 0.5 < P < 1 | Rolling a number greater than 2 on a die (≈ 0.67) |
| Certain | 1 | It being daytime somewhere on Earth right now |
Converting Between Representations
Probability values can be written three ways, always be comfortable switching between them:
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 4/52 = 1/13 | 0.077 | 7.7% |
To convert: fraction → decimal: divide numerator by denominator. Decimal → percentage: multiply by 100.
Placing Events on the Scale, Worked Examples
Calculate the probability first, then decide where it sits:
- $P(\text{roll 7 on die}) = 0$ → Impossible (7 is not on the die)
- $P(\text{draw Ace from deck}) = \dfrac{4}{52} \approx 0.077$ → Unlikely (between 0 and 0.5)
- $P(\text{flip heads}) = \dfrac{1}{2} = 0.5$ → Even chance
- $P(\text{roll} > 2 \text{ on die}) = \dfrac{4}{6} \approx 0.67$ → Likely (between 0.5 and 1)
- $P(\text{rain somewhere in Australia today}) = 1$ → Certain
Comparing Probabilities
When $P(A) > P(B)$, event A is more likely than event B. The probability scale is just a number line, bigger numbers sit further to the right.
Example, ordering by likelihood:
$P(\text{roll} = 1) = \dfrac{1}{6} \approx 0.17 \qquad P(\text{roll} > 2) = \dfrac{4}{6} \approx 0.67$
So rolling $> 2$ is much more likely than rolling exactly 1. In order from least to most likely:
Roll 7 (impossible, $P=0$) < Draw Ace ($P \approx 0.08$) < Roll = 1 ($P \approx 0.17$) < Flip heads ($P = 0.5$) < Roll $> 2$ ($P \approx 0.67$) < Rain in AU today ($P = 1$)
Common Pitfalls
- Confusing "unlikely" with "impossible", unlikely events CAN and DO happen
- Thinking P = 0.5 means "I don't know", it means equal chance, which is precise information
- Placing events on the scale without first calculating their probability value
- Comparing fractions without converting to decimals first, 3/7 vs 5/12 is hard to compare without converting
- Forgetting that percentages need to be divided by 100 to get a probability value (e.g. 70% = 0.7)
Copy This Into Your Book
The Probability Scale:
$$0 \longleftarrow \text{impossible} \quad \text{unlikely} \quad \underbrace{0.5}_{\text{even chance}} \quad \text{likely} \quad \text{certain} \longrightarrow 1$$
Converting: fraction $\to$ decimal (÷ bottom by top), decimal $\to$ % (× 100).
$P(A) > P(B)$ means A is more likely than B. Order events by their numerical values.
An event has probability 0. This means the event is:
A probability of 0.5 on the scale means the event is:
Which of these events is the least likely?
An event has probability 0.8. Which label best describes it on the probability scale?
Which of the following events is certain?
Q6. Five events have the following probabilities: 0, 0.1, 0.5, 0.8, 1.
(a) Draw a number line from 0 to 1 and place all five values on it.
(b) Label each with the correct scale descriptor (impossible / unlikely / even chance / likely / certain).
(c) Which two values are complements of each other? Explain why.
Q7. Convert the following probabilities between all three forms (fraction, decimal, percentage):
(a) $\dfrac{3}{4}$ (b) 0.6 (c) 35% (d) $\dfrac{7}{20}$ (e) 0.08
For each one, also state whether the event is impossible, unlikely, even chance, likely, or certain.
Q8. A bag contains some coloured marbles. Consider these four events:
Event A: drawing a red marble. Event B: drawing a marble that is NOT blue. Event C: drawing a marble that is purple (when there are no purple marbles). Event D: drawing any marble at all.
(a) Write the probability of Event C and classify it on the probability scale.
(b) Write the probability of Event D and classify it.
(c) If P(red) = 3/10, classify Event A and find P(not red).
(d) If P(blue) = 2/10, find P(not blue) and classify Event B.
Show Answers
Q6
(a) Number line: 0,,, 0.1,,, 0.5,,, 0.8,,, 1
(b) 0 = impossible; 0.1 = unlikely; 0.5 = even chance; 0.8 = likely; 1 = certain.
(c) 0.1 and 0.9 are complements (0.1 + 0.9 = 1), but 0.9 is not listed. From the given list, 0 and 1 are complements, if an event is impossible, its complement (the event NOT happening) is certain.
Q7
(a) 3/4 = 0.75 = 75%, likely.
(b) 0.6 = 3/5 = 60%, likely.
(c) 35% = 0.35 = 7/20, unlikely.
(d) 7/20 = 0.35 = 35%, unlikely.
(e) 0.08 = 2/25 = 8%, unlikely.
Q8
(a) P(purple) = 0, impossible (no purple marbles in the bag).
(b) P(any marble) = 1, certain (every draw will produce some marble).
(c) P(red) = 3/10 = 0.3, unlikely. P(not red) = 1 − 3/10 = 7/10 = 0.7.
(d) P(blue) = 2/10 = 0.2. P(not blue) = 1 − 0.2 = 0.8, likely.
Stretch Challenge
A spinner is divided into 8 equal sections, each labelled with a number. You want to design it so that:
• $P(\text{even number}) = 0.5$
• $P(\text{number} = 1) = 0.125$
• $P(\text{number} = 8) = 0.25$
(a) How many sections show the number 1? Justify using the probability.
(b) How many sections show the number 8?
(c) P(even) = 0.5 means 4 sections show even numbers. Two sections show 8 (which is even). What other even numbers could fill the remaining 2 even sections?
(d) Is there more than one valid design for this spinner? Explain.
(e) What is $P(\text{not } 8)$ on your spinner?