Counting and probability: QCE Math Methods Unit 1 Year 11

What this dot point is asking

QCAA wants Year 11 students to apply counting techniques (the multiplication principle, permutations and combinations) to compute probabilities, to use set notation fluently, and to apply conditional probability, the addition and multiplication rules, and independence. These ideas underpin the discrete-probability and distribution work later in the course, so secure technique here pays off across the subject.

Counting principles

The multiplication principle says that if a first task can be done in $m$ ways and a second independently in $n$ ways, the two together can be done in $m \times n$ ways. This generalises to any number of stages.

A permutation is an arrangement where order matters. The number of ordered selections of $k$ objects from $n$ distinct objects is

A combination is a selection where order does not matter. The number of unordered selections of $k$ from $n$ is

The factor $k!$ in the denominator removes the $k!$ orderings that a permutation would count separately.

Set notation

Probability is built on sets. The sample space $S$ is the set of all outcomes; an event $A$ is a subset of $S$. The union $A \cup B$ contains outcomes in $A$ or $B$ (or both); the intersection $A \cap B$ contains outcomes in both; and the complement $A'$ contains outcomes not in $A$. A Venn diagram is often the fastest way to organise overlapping events before applying a rule.

Simple probability

For equally likely outcomes, the probability of an event is the proportion of the sample space it occupies:

Every probability satisfies $0 \leq P(A) \leq 1$, and the complement rule $P(A') = 1 - P(A)$ is often the quickest route, especially for "at least one" questions.

The addition rule

For any two events,

The overlap is subtracted so it is not counted twice. When the events are mutually exclusive they cannot both occur, so $P(A \cap B) = 0$ and the rule simplifies to $P(A \cup B) = P(A) + P(B)$.

Conditional probability and the multiplication rule

The conditional probability of $A$ given that $B$ has occurred is

Rearranging gives the multiplication rule $P(A \cap B) = P(A \mid B)\,P(B)$. Two events are independent when the occurrence of one does not change the probability of the other, that is $P(A \mid B) = P(A)$, in which case $P(A \cap B) = P(A)\,P(B)$. Sampling without replacement is dependent, because removing an item changes the composition of what remains.

In one sentence

Counting combines the multiplication principle with permutations ($P(n,k) = \tfrac{n!}{(n-k)!}$, order matters) and combinations ($\binom{n}{k} = \tfrac{n!}{k!(n-k)!}$, order does not), and probability is governed by $P(A) = \tfrac{|A|}{|S|}$, the addition rule, conditional probability $P(A \mid B) = \tfrac{P(A \cap B)}{P(B)}$, and independence.