Sample spaces and probability
In Year 12 Mathematical Methods (Queensland, QCAA), the sample space is the set of all possible outcomes and an event is a subset of them. For equally likely outcomes, \(P(A)=\dfrac{\text{favourable}}{\text{total}}\). The complement, addition and mutually exclusive rules combine event probabilities, Venn diagrams and two-way tables organise the counts, and relative frequency estimates a probability from data.
A chance experiment (such as rolling a die) has a set of possible outcomes. The sample space is the set of all of them, and an event is any subset of the sample space — for example, 'rolling an even number' is the event \(\{2,4,6\}\).
When every outcome is equally likely, the probability of an event is
\(P(A)=\dfrac{\text{number of outcomes in }A}{\text{total number of outcomes}},\qquad 0\le P(A)\le 1.\)
The complement \(A'\) is 'not \(A\)'. Two events are mutually exclusive if they cannot both occur. Set notation writes \(A\cap B\) for 'A and B' (the intersection) and \(A\cup B\) for 'A or B' (the union), which a Venn diagram pictures as overlapping circles.
Probability of an event (equally likely outcomes):
The complement and addition rules:
Mutually exclusive events (no overlap) and relative frequency:
How to find a probability
- List the sample space. Identify all possible outcomes and how many there are; check whether they are equally likely.
- Count the favourable outcomes. For an equally likely space, \(P(A)=\dfrac{\text{favourable}}{\text{total}}\).
- Use a rule if events combine. For 'not \(A\)' use \(P(A')=1-P(A)\); for '\(A\) or \(B\)' use \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\), dropping the overlap only when the events are mutually exclusive.
- Organise with a diagram or table. A Venn diagram or two-way table lays out the counts so each region or cell can be read directly.
Favourable outcomes over total outcomes.
| \(P(\text{red})\) | \(=\) | \(\dfrac{3}{3+7}=\dfrac{3}{10}\) |
There are \(2\) \(E\)'s among \(5\) letters, so use the complement.
| \(P(\text{not }E)\) | \(=\) | \(1-\dfrac{2}{5}\) |
| \(=\) | \(\dfrac{3}{5}\) |
Subtract the overlap once so it is not counted twice.
| \(P(A\cup B)\) | \(=\) | \(0.55+0.30-0.15\) |
| \(=\) | \(0.70\) |
Relative frequency is the count divided by the number of trials.
| \(P(6)\) | \(\approx\) | \(\dfrac{42}{250}=0.168\) |
Common pitfalls
Frequently asked questions
How do you find the probability of an event?
For equally likely outcomes, divide the favourable outcomes by the total. A bag of \(3\) red and \(7\) yellow gives \(P(\text{red})=\dfrac{3}{10}=0.3\). Every probability lies between \(0\) and \(1\).
What is the complement rule?
\(A'\) is 'not \(A\)', and \(P(A')=1-P(A)\). If \(P(6)=\dfrac{1}{6}\), then \(P(\text{not }6)=\dfrac{5}{6}\). It is handy for 'at least one' problems.
What is the addition rule for probability?
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\). For \(P(A)=0.55\), \(P(B)=0.30\), \(P(A\cap B)=0.15\), you get \(P(A\cup B)=0.70\).
What does mutually exclusive mean?
Two events that cannot both happen, so \(P(A\cap B)=0\); then \(P(A\cup B)=P(A)+P(B)\). Rolling a \(5\) and a \(6\) on one die are mutually exclusive.
What is relative frequency?
The number of times an event happens divided by the number of trials. \(250\) rolls with \(42\) sixes give a relative frequency of \(\dfrac{42}{250}=0.168\).
How do Venn diagrams and two-way tables help?
They organise the counts — a Venn diagram into only \(A\), the overlap, only \(B\) and outside; a two-way table by two attributes — so a probability is the relevant count over the total.