The Probability Scale
Get to know the probability scale for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). Every probability is a number between 0 (impossible) and 1 (certain) that measures how likely an event is.
You will place chances on the scale from 0 to 1, describe them in words such as unlikely, even chance and likely, and write the same probability as a fraction, decimal or percentage — so you can compare chances and make sense of everyday statements like a 70% chance of rain.
Theory
The probability of an event is a number from 0 (impossible) to 1 (certain) that measures how likely it is to happen. This Year 12 Standard 1 (NSW) guide shows how to read the probability scale, describe chance in words, and convert between fractions, decimals and percentages.
The probability \(P\) of an event is a number from \(0\) to \(1\) (or \(0\%\) to \(100\%\)) that measures how likely it is. A value below \(0\) or above \(1\) cannot be a probability.
In words: \(0\) is impossible; less than \(0.5\) is unlikely; \(0.5\) is an even chance; more than \(0.5\) is likely; \(1\) is certain. The larger the probability, the more likely the event.
A probability can be written as a fraction, a decimal or a percentage, e.g. \(\dfrac{3}{4}=0.75=75\%\), which is likely.
Every probability lies on the scale from \(0\) to \(1\):
Converting between forms:
How to place and compare probabilities
- Convert each probability to a decimal (divide a fraction; divide a percentage by \(100\)).
- Check it lies from \(0\) to \(1\); otherwise it cannot be a probability.
- Describe it in words: impossible, unlikely, even chance, likely or certain.
- Compare the decimals — the larger one is the more likely event.
Write each as a decimal, then place it on the scale.
| \(\text{(i) no } 9 \text{ on a die}\) | \(\Rightarrow\) | \(P=0,\ \text{impossible}\) |
| \(\text{(ii) } 15\% = 0.15 < 0.5\) | \(\Rightarrow\) | \(\text{unlikely}\) |
| \(\text{(iii) } P=\dfrac{1}{2}=0.5\) | \(\Rightarrow\) | \(\text{even chance}\) |
| \(\text{(iv) } 0.97 \text{ close to } 1\) | \(\Rightarrow\) | \(\text{very likely}\) |
(i) impossible, (ii) unlikely, (iii) even chance, (iv) very likely.
Divide for a decimal, multiply by \(100\%\) for a percentage, and write over \(100\) for a fraction.
| \(\text{(i) } \dfrac{3}{8}\) | \(=\) | \(3\div 8 = 0.375\) |
| \(0.375\) | \(=\) | \(0.375\times 100\% = 37.5\%\) |
| \(\text{(ii) } 35\%\) | \(=\) | \(\dfrac{35}{100} = \dfrac{35\div 5}{100\div 5} = \dfrac{7}{20}\) |
(i) \(0.375\), \(37.5\%\). (ii) \(\dfrac{7}{20}\).
Each small division is \(0.1\).
| \(\text{(i) } A,\ B,\ C\) | \(=\) | \(0.3,\ 0.7,\ 0.95\) |
| \(\text{(ii) } C \text{ closest to } 1\) | \(\Rightarrow\) | \(\text{most likely}\) |
| \(\text{(iii) } 0.7\times 100\%\) | \(=\) | \(70\%\) |
\(C\) is halfway between \(0.9\) and \(1\), so \(C=0.95\). (i) \(0.3\), \(0.7\), \(0.95\). (ii) \(C\). (iii) \(70\%\).
A probability must be from \(0\) to \(1\); change everything to decimals to compare.
| \(\text{(i) } 1.05\) | \(>\) | \(1\) |
| \(-0.2\) | \(<\) | \(0\) |
| \(110\%\) | \(=\) | \(1.1 > 1\) |
| \(\text{(ii) } \dfrac{2}{5}\) | \(=\) | \(0.4\) |
| \(45\%\) | \(=\) | \(0.45\) |
| \(\text{order}\) | \(:\) | \(0.35,\ 0.4,\ 0.45,\ 0.6\) |
(i) \(1.05\), \(-0.2\) and \(110\%\) cannot be probabilities. (ii) \(\mathbf{0.35,\ \dfrac{2}{5},\ 45\%,\ 0.6}\).
Common pitfalls
Frequently asked questions
What is the probability scale?
The probability scale runs from 0 to 1. A probability of 0 means the event is impossible, 0.5 means an even chance, and 1 means the event is certain. The larger the probability, the more likely the event.
Can a probability be greater than 1?
No. Every probability is between 0 and 1 inclusive, or 0% to 100%. Values such as 1.3, 5/4, 120% or -0.2 cannot be probabilities.
What words describe probability?
0 is impossible, less than 0.5 is unlikely, 0.5 is an even chance, more than 0.5 is likely, and 1 is certain. Very close to 0 is very unlikely and very close to 1 is very likely.
How do you change a fraction into a percentage?
Divide the top by the bottom to get a decimal, then multiply by 100%. For example, 3/8 = 3 divided by 8 = 0.375, which is 37.5%.
Is a very unlikely event impossible?
No. A probability of 0.002 is tiny, but the event can still happen. Only a probability of exactly 0 means impossible.