Range of Probabilities
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Theory
Every probability is a number from \(0\) to \(1\) (or \(0\%\) to \(100\%\)). \(0\) means impossible, \(\dfrac{1}{2}\) is an even chance and \(1\) means certain. This Year 7 maths page places events on the probability scale, describes them in words and compares probabilities in different forms.
Every probability is from \(0\) to \(1\): \(0 \le P(E) \le 1\). As a percentage it is from \(0\%\) to \(100\%\).
\(P = 0\): the event is impossible (no favourable outcomes). \(P = 1\): it is certain (every outcome is favourable). \(P = \dfrac{1}{2}\): an even chance.
Between \(0\) and \(\dfrac{1}{2}\) the event is unlikely; between \(\dfrac{1}{2}\) and \(1\) it is likely. To place a probability, compare it with \(\dfrac{1}{2}\). Very close to \(0\) is very unlikely; very close to \(1\) is very likely.
To compare probabilities in different forms, write each one as a decimal. The bigger the number, the more likely the event.
The probabilities of all the outcomes add to \(1\). On a fair die each face has \(P = \dfrac{1}{6}\), and \(\dfrac{6}{6} = 1\).
The scale in words:
| Probability | Words |
|---|---|
| \(0\) | impossible |
| between \(0\) and \(\dfrac{1}{2}\) | unlikely |
| \(\dfrac{1}{2}\) | even chance |
| between \(\dfrac{1}{2}\) and \(1\) | likely |
| \(1\) | certain |
How to describe the chance of an event
- Find the probability (favourable over total).
- Compare it with \(0\), \(\dfrac{1}{2}\) and \(1\).
- Choose the word: impossible, unlikely, even chance, likely or certain.
How to order probabilities
- Write each one as a decimal.
- Order the decimals from smallest (least likely) to largest (most likely).
(i) \(P(\text{greater than } 2)\)
(ii) \(P(\text{greater than } 6)\)
(i) \(\{3, 4, 5, 6\}\): \(\dfrac{4}{6} = \dfrac{2}{3}\) (\(4\) chances in \(6\)), likely
(ii) No number is greater than \(6\): \(\dfrac{0}{6} = 0\), impossible
(i) \(P(\text{red})\)
(ii) \(P(\text{white})\)
(iii) \(P(\text{red or white})\)
(iv) \(P(\text{black})\)
(i) \(\dfrac{10}{11}\): very likely
(ii) \(\dfrac{1}{11}\): very unlikely
(iii) \(\dfrac{11}{11} = 1\): certain
(iv) \(\dfrac{0}{11} = 0\): impossible
(i) \(P(\text{blue})\)
(ii) \(P(\text{yellow})\)
(iii) \(P(\text{purple})\)
Compare with \(\dfrac{1}{2} = \dfrac{4}{8}\).
(i) \(\dfrac{5}{8}\), likely
(ii) \(\dfrac{3}{8}\), unlikely
(iii) \(0\), impossible
\(0.45\): yes. \(\dfrac{7}{6} = 1\dfrac{1}{6}\) is more than \(1\): no. \(-0.2\) is less than \(0\): no. \(100\% = 1\): yes (certain).
| \(\dfrac{3}{10}\) | \(=\) | \(0.3\) |
| \(\dfrac{45}{100}\) | \(=\) | \(0.45\) |
| \(\dfrac{1}{4}\) | \(=\) | \(0.25\) |
\(0.25 < 0.3 < 0.45 < 0.6\): Z, X, Y, W. Only W is above \(0.5\), so only W is likely.
(i) Show that these add to \(1\).
(ii) Which colour is most likely? Describe its chance.
(iii) Could the spinner also land on yellow?
| \(0.35 + 0.25 + 0.4\) | \(=\) | \(1\) |
(i) \(\dfrac{1}{4} = 0.25\) and \(40\% = 0.4\); the sum is \(1\).
(ii) Green, \(0.4\). It is below \(0.5\), so even green is unlikely.
(iii) No: the three colours already make \(1\), so \(P(\text{yellow}) = 0\).
Common pitfalls
Frequently asked questions
What is the range of probabilities?
From \(0\) (impossible) to \(1\) (certain), or \(0\%\) to \(100\%\).
What probability is an even chance?
\(\dfrac{1}{2}\), \(0.5\) or \(50\%\).
Can a probability be more than 1?
No. \(1\) means certain, and nothing is more than certain.
How do you compare probabilities?
Write them all as decimals, then compare.
What do the probabilities of all outcomes add to?
\(1\).