Two-Stage Events - Arrays
Use arrays to handle two-stage events in NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). An array is a table that sets out every equally likely outcome of an experiment done in two parts, such as rolling two dice or tossing a coin and rolling a die.
You will set out equally likely outcomes in an array, count the total as rows times columns, find probabilities such as the sum of two dice, and handle draws with replacement — a clear way to solve problems from board games and spinner games.
Theory
An array sets out every outcome of a two-stage event in a table. This Year 12 Standard 1 (NSW) guide shows how to build arrays for coins, dice and spinners, count the cells, and find probabilities such as the sum of two dice and draws with replacement.
A two-stage event is an experiment done in two parts: two coins, a coin and a die, two spins, or two draws with replacement. An array (a table) lists every outcome of a two-stage event.
Put the first stage down the side and the second stage across the top. Each cell holds one outcome, such as \(\text{H}4\), the pair \((2,5)\), or the sum of the two numbers. The total number of outcomes is rows \(\times\) columns.
When each stage is fair, every cell is equally likely, so a probability is the number of favourable cells over the total number of cells. If an item is drawn and replaced before the second draw, the second stage is the same as the first, so doubles like \((2,2)\) are included.
The number of cells is the two stage sizes multiplied:
Count the cells that fit the event, then:
How to use an array
- Draw the array: first stage down the side, second stage across the top, and fill in every cell (the pair or the sum).
- Count the total cells (rows \(\times\) columns) and shade the cells that fit the event.
- Divide favourable cells by total cells and simplify.
Coin down the side, spinner across the top: \(2\times4=8\) equally likely cells.
| \(A\) | \(B\) | \(C\) | \(D\) | |
|---|---|---|---|---|
| \(H\) | \(\text{HA}\) | \(\text{HB}\) | \(\text{HC}\) | \(\text{HD}\) |
| \(T\) | \(\text{TA}\) | \(\text{TB}\) | \(\text{TC}\) | \(\text{TD}\) |
| \(\text{favourable}\) | \(=\) | \(\text{TB, TC}\Rightarrow 2\) |
| \(P(\text{tail and B or C})\) | \(=\) | \(\dfrac{2}{8}=\dfrac{1}{4}\) |
\(P=\dfrac{1}{4}\).
X down the side, Y across the top: \(3\times4=12\) cells.
| \(+\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
| \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
| \(3\) | \(4\) | \(5\) | \(6\) | \(7\) |
| \(\text{sum}=5\) | \(:\) | \((1,4),(2,3),(3,2)\Rightarrow 3\) |
| \(P(\text{sum}=5)\) | \(=\) | \(\dfrac{3}{12}=\dfrac{1}{4}\) |
\(P(\text{sum}=5)=\dfrac{1}{4}\).
The counter is replaced, so all \(4\times4=16\) cells are equally likely.
| \(+\) | \(2\) | \(4\) | \(5\) | \(7\) |
|---|---|---|---|---|
| \(2\) | \(4\) | \(6\) | \(7\) | \(9\) |
| \(4\) | \(6\) | \(8\) | \(9\) | \(11\) |
| \(5\) | \(7\) | \(9\) | \(10\) | \(12\) |
| \(7\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \(\text{sum}>10\) | \(:\) | \(11,12,11,12,14\Rightarrow 5\) |
| \(P(\text{sum}>10)\) | \(=\) | \(\dfrac{5}{16}\) |
\(P(\text{sum}>10)=\dfrac{5}{16}\) (note \(5+5=10\) is not more than \(10\)).
Count the cells for each event.
| \(\text{(i) total}\) | \(=\) | \(6\times6=36\) |
| \(\text{(ii) sum}=4\) | \(:\) | \((1,3),(2,2),(3,1)\Rightarrow 3\) |
| \(P\) | \(=\) | \(\dfrac{3}{36}=\dfrac{1}{12}\) |
| \(\text{(iii) sums }2,3,4,5\) | \(:\) | \(1+2+3+4=10\) |
| \(P\) | \(=\) | \(\dfrac{10}{36}=\dfrac{5}{18}\) |
(i) \(36\) (ii) \(\dfrac{1}{12}\) (iii) \(\dfrac{5}{18}\)
Common pitfalls
Frequently asked questions
What is an array in probability?
An array is a table that lists every outcome of a two-stage experiment. The first stage goes down the side, the second stage goes across the top, and each cell is one outcome.
How many outcomes are there when you roll two dice?
There are 6 times 6 = 36 outcomes. Each die has 6 faces, and every face of the first die can pair with every face of the second.
What is the most likely sum when rolling two dice?
A sum of 7 is the most likely. It appears in 6 of the 36 cells of the sum array, so its probability is 6 over 36, which is 1 sixth.
How do you find the total number of outcomes for two stages?
Multiply the number of outcomes in each stage. A coin and a die give 2 times 6 = 12 outcomes, not 2 plus 6.
Do (2, 5) and (5, 2) count as different outcomes?
Yes. In an array the first number comes from the first stage and the second from the second stage, so (2, 5) and (5, 2) are two different cells and both are counted.
What does with replacement mean?
With replacement means the first item is put back before the second draw. The second draw has the same choices as the first, so doubles such as (2, 2) are possible.