Probability
Every question with a fully worked solution.
- Probability - Video - Basic Probability Watch
Theory
The probability of an event tells you how likely it is. When all the outcomes are equally likely, \(P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}\). This Year 7 maths page shows how to count the outcomes, handle “or” and “not”, and write a probability as a fraction, decimal or percentage.
An experiment is a situation involving chance that leads to outcomes. A trial is one run of the experiment, and an outcome is a result. An event consists of one or more outcomes. \(P(E)\) means the probability of an event \(E\).
\(P(E) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}\) when the outcomes are equally likely (a fair die, equal sectors, or a choice made at random). This is the theoretical (expected) probability.
Step 1. Count all the equally likely outcomes: every card, marble or equal sector, including repeats. (The bag in Diagrams has \(3\) colours but \(8\) marbles.) Step 2. Count the favourable outcomes. Step 3. Write favourable over total and simplify (divide by the HCF).
For an “or” event, count each favourable outcome only once. On a die, “even or greater than \(4\)” is \(\{2, 4, 5, 6\}\): \(4\) outcomes, not \(3 + 2 = 5\), because \(6\) is in both. A “not” event uses every outcome that is not in the event.
A probability can be a fraction, a decimal or a percentage. Make the denominator \(100\): \(\dfrac{9}{20} = \dfrac{45}{100} = 0.45 = 45\%\).
The rule:
How to find a probability
- Count the total number of equally likely outcomes.
- Count the favourable outcomes (each one once).
- Write favourable over total.
- Simplify, or change to a decimal or percentage if asked.
(i) \(P(\text{red})\)
(ii) \(P(\text{not white})\)
(iii) \(P(\text{blue, red or white})\)
(iv) \(P(\text{green})\)
| \(4 + 3 + 2\) | \(=\) | \(9\) |
| \(\dfrac{3}{9}\) | \(=\) | \(\dfrac{1}{3}\) |
| \(\dfrac{3 + 4}{9}\) | \(=\) | \(\dfrac{7}{9}\) |
| \(\dfrac{4 + 3 + 2}{9}\) | \(=\) | \(1\) |
(i) \(\dfrac{1}{3}\)
(ii) not white means red or blue: \(\dfrac{7}{9}\)
(iii) \(1\) (certain)
(iv) \(\dfrac{0}{9} = 0\) (no green counters)
(i) \(P(5)\)
(ii) \(P(\text{prime number})\)
(iii) \(P(\text{not an even number})\)
(i) One \(5\) among \(6\) numbers: \(\dfrac{1}{6}\)
(ii) The primes are \(2, 3, 5\): \(\dfrac{3}{6} = \dfrac{1}{2}\)
(iii) Not even means \(1, 3, 5\): \(\dfrac{3}{6} = \dfrac{1}{2}\)
(i) \(P(7)\)
(ii) \(P(\text{even})\)
(iii) \(P(\text{less than } 4)\)
(i) \(\dfrac{1}{10}\)
(ii) \(2, 4, 6, 8, 10\): \(\dfrac{5}{10} = \dfrac{1}{2}\)
(iii) \(1, 2, 3\) (not \(4\)): \(\dfrac{3}{10}\)
(i) \(P(\text{Anzac})\)
(ii) \(P(\text{Anzac or shortbread})\)
| \(5 + 4 + 3\) | \(=\) | \(12\) |
| \(\dfrac{5 + 3}{12}\) | \(=\) | \(\dfrac{8}{12} = \dfrac{2}{3}\) |
(i) \(\dfrac{5}{12}\)
(ii) \(\dfrac{2}{3}\)
| \(11 + 8 + 6\) | \(=\) | \(25\) |
| \(\dfrac{8}{25}\) | \(=\) | \(\dfrac{32}{100}\) |
\(\dfrac{8}{25}\), \(0.32\), \(32\%\)
(i) How many tickets were sold?
(ii) How many more tickets must be sold to make \(P(\text{Mia wins}) = \dfrac{1}{10}\)?
| \(\dfrac{1}{8}\) | \(=\) | \(\dfrac{5}{40}\) |
| \(\dfrac{1}{10}\) | \(=\) | \(\dfrac{5}{50}\) |
| \(50 - 40\) | \(=\) | \(10\) |
(i) \(40\) tickets
(ii) \(10\) more tickets
| \(\dfrac{1}{4}\) | \(=\) | \(\dfrac{3}{12}\) |
| \(12 - 9\) | \(=\) | \(3\) |
The total must be \(12\), and there are \(9\) now, so add \(3\) lemon juices. Check: \(\dfrac{3}{12} = \dfrac{1}{4}\).
Common pitfalls
Frequently asked questions
How do you calculate a probability?
Divide the number of favourable outcomes by the total number of equally likely outcomes.
What is the probability of rolling a 2 on a die?
\(\dfrac{1}{6}\): one favourable outcome out of six.
Can a probability be written as a percentage?
Yes. \(\dfrac{9}{20} = 0.45 = 45\%\).
What is the probability of something that cannot happen?
\(0\). For example, a green counter from a bag with none.
What does at random mean?
Every object has the same chance of being chosen.