Probability
Every question with a fully worked solution.
- Probability - Video - Probability Watch
Theory
The probability of an event is the number of favourable outcomes out of the total number of equally likely outcomes. A bag of \(4\) blue, \(3\) red and \(2\) white counters gives \(P(\text{red}) = \dfrac{3}{9} = \dfrac{1}{3}\). This Year 8 maths page shows how.
Words. An experiment is a situation involving chance; a trial is one run of it; an outcome is a possible result. The sample space is the set of all possible outcomes, and an event is one or more of the outcomes.
Equally likely outcomes each have the same chance, such as the faces of a fair die or equal sectors of a spinner. Selecting at random means no outcome is favoured.
The formula. \(P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}\). Count every equally likely outcome (each counter or card, repeats included), not each colour.
The scale. A probability runs from \(0\) (impossible) to \(1\) (certain); \(\dfrac{1}{2}\) is an even chance. It can be written as a fraction, a decimal or a percentage, and the probabilities of all the outcomes add to \(1\).
‘Or’ events. Count each outcome once: \(n(\text{A or B}) = n(\text{A}) + n(\text{B}) - n(\text{both})\).
Comparing. More favourable outcomes does not always mean a better chance: compare the fractions with a common denominator.
Both ways. Number in a group \(=\) probability \(\times\) total. To find a total, build an equivalent fraction with the numerator you know. A game is fair when the players have equal probabilities of scoring.
A tub of \(25\) counters: \(10\) red, \(9\) blue, \(6\) yellow.
| Event | Fraction | Decimal | Per cent |
|---|---|---|---|
| red | \(\dfrac{10}{25} = \dfrac{2}{5}\) | \(0.4\) | \(40\%\) |
| blue | \(\dfrac{9}{25}\) | \(0.36\) | \(36\%\) |
| yellow | \(\dfrac{6}{25}\) | \(0.24\) | \(24\%\) |
| red or yellow | \(\dfrac{16}{25}\) | \(0.64\) | \(64\%\) |
Finding a probability
- List or count the sample space of equally likely outcomes.
- Count the favourable outcomes (shared outcomes once).
- Write favourable over total and simplify; convert to a decimal or percentage if asked.
Working backwards
- Number \(=\) probability \(\times\) total.
- Total: make an equivalent fraction whose numerator is the known count.
(i) selecting a red counter
(ii) not selecting a white counter
(iii) selecting a blue, red or white counter
(iv) selecting a green counter.
| \(P(\text{red})\) | \(=\) | \(\dfrac{3}{9} = \dfrac{1}{3}\) |
| \(P(\text{not white})\) | \(=\) | \(\dfrac{3 + 4}{9} = \dfrac{7}{9}\) |
| \(P(\text{any})\) | \(=\) | \(\dfrac{9}{9} = 1\) |
| \(P(\text{green})\) | \(=\) | \(\dfrac{0}{9} = 0\) |
(i) \(\dfrac{1}{3}\)
(ii) \(\dfrac{7}{9}\)
(iii) \(1\) (certain)
(iv) \(0\) (impossible)
(i) rolling a \(4\)
(ii) rolling a prime number
(iii) not rolling an even number.
There are \(6\) equally likely outcomes. The primes are \(2, 3, 5\); the odd numbers are \(1, 3, 5\).
| \(P(4)\) | \(=\) | \(\dfrac{1}{6}\) |
| \(P(\text{prime})\) | \(=\) | \(\dfrac{3}{6} = \dfrac{1}{2}\) |
| \(P(\text{not even})\) | \(=\) | \(\dfrac{3}{6} = \dfrac{1}{2}\) |
(i) \(\dfrac{1}{6}\)
(ii) \(\dfrac{1}{2}\)
(iii) \(\dfrac{1}{2}\)
(i) \(P(\text{rock})\) in simplest form
(ii) \(P(\text{hip hop or country})\) as a percentage.
| \(P(\text{rock})\) | \(=\) | \(\dfrac{6}{30} = \dfrac{1}{5}\) |
| \(9 + 3\) | \(=\) | \(12\) |
| \(P\) | \(=\) | \(\dfrac{12}{30} = \dfrac{40}{100}\) |
(i) \(\dfrac{1}{5}\)
(ii) \(40\%\): \(12\) songs are hip hop or country, and no song is in both groups.
Multiples of \(3\): \(3, 6, 9, 12, 15, 18\). Factors of \(18\): \(1, 2, 3, 6, 9, 18\). In both: \(3, 6, 9, 18\).
| \(6 + 6 - 4\) | \(=\) | \(8\) |
| \(P\) | \(=\) | \(\dfrac{8}{18} = \dfrac{4}{9}\) |
\(\dfrac{4}{9}\)
| \(P(\text{Ivy})\) | \(=\) | \(\dfrac{3}{15} = \dfrac{1}{5}\) |
| \(P(\text{Max})\) | \(=\) | \(\dfrac{3}{15} = \dfrac{1}{5}\) |
Ivy has \(\{4, 8, 12\}\) and Max has \(\{1, 2, 3\}\). The probabilities are equal, so the game is fair.
(i) How many are strawberry?
(ii) Find \(P(\text{cola})\).
(iii) How many cola lollies must be taken out so that \(P(\text{lime}) = \dfrac{2}{5}\)?
| \(0.4 \times 60\) | \(=\) | \(24\) |
| \(60 - 24 - 18\) | \(=\) | \(18\) |
| \(P(\text{cola})\) | \(=\) | \(\dfrac{18}{60} = \dfrac{3}{10}\) |
| \(\dfrac{2}{5}\) | \(=\) | \(\dfrac{18}{45}\) |
(i) \(24\)
(ii) \(\dfrac{3}{10}\)
(iii) The \(18\) lime stay, so the total must be \(45\): take out \(15\) cola.
Common pitfalls
Frequently asked questions
How do you calculate probability?
Divide the number of favourable outcomes by the total number of equally likely outcomes: \(P(\text{a } 6) = \dfrac{1}{6}\) on a fair die.
What is a sample space?
The set of all possible outcomes of an experiment.
Can a probability be greater than 1?
No. Probabilities run from 0 (impossible) to 1 (certain).
How do I find the probability of A or B?
Count the outcomes in A or B, each one only once, and divide by the total.
What makes a game fair?
Each player has the same probability of winning or scoring.