Multistage Events — Arrays and Tree Diagrams

Before you can calculate a probability, you need to list all possible outcomes. For two-stage experiments, an array (a grid) does this mechanically — rows for one stage, columns for the other, every cell is one outcome.

How to build and use an array:

• Set up the grid: Write one stage's outcomes across the top (columns) and the other stage's outcomes down the side (rows). Each cell is an outcome pair.
• Example — die and coin: Roll a die (columns 1–6) and flip a coin (rows H, T). The array is 2 × 6 = 12 cells; each cell is e.g. (H,1), (H,2), …, (T,6).
• Finding probability from an array: Count cells matching your event and divide by total cells.

An array (two-way table) lists all outcomes of a two-stage experiment in a grid. Rows represent one stage, columns represent the other. Each cell is one equally-likely outcome. Count favourable cells to find probability.

Pause — copy the two-way array structure (rows = one stage outcomes; columns = other stage outcomes; each cell = one combined outcome) and the probability counting rule: P(event) = number of favourable cells ÷ total number of cells into your book.

A two-way array works well when both stages have the same set of outcomes (e.g., two dice) and the grid structure is compact. When the two stages have different numbers of outcomes, or there are more than two stages, a tree diagram is more flexible: it branches at each stage regardless of how many outcomes there are, and each path through the tree represents one complete outcome.

Arrays work for two stages — but tree diagrams extend to any number of stages and handle changing probabilities (as in "without replacement") naturally.

A tree diagram starts from a single point and branches at each stage. Every possible sequence of outcomes is one complete path from left to right.

How to draw a tree diagram:

• Draw a dot (starting point). From it, draw one branch for each outcome at stage 1, labelled with the outcome.
• From each stage-1 endpoint, draw branches for each stage-2 outcome, also labelled.
• Reading outcomes: Each path (left to right) is one combined outcome. List all paths to get the sample space.
• Counting: $n(S) =$ (number of stage-1 branches) × (number of stage-2 branches per branch).

A tree diagram shows sequential outcomes by branching at each stage. Each branch is labelled with the outcome and its probability. To find the probability of a path, multiply the branch probabilities along that path.

Pause — copy the tree diagram multiplication rule: multiply branch probabilities along a single path to get the probability of that complete sequence — and note each path must be labelled with its outcome and probability into your book.

A tree diagram labels each branch with its probability, and following one complete path gives one outcome. To find the probability of a specific path, multiply all the branch probabilities along it: P(A and B) = P(A) × P(B|A). When an event can occur via multiple paths, add the probabilities of all valid paths together.

A tree diagram shows outcomes; a probability tree shows both outcomes AND their probabilities. Once you have a probability tree, calculating P(any path) is mechanical.

A probability tree adds the probability of each branch. To find the probability of any complete path (a specific sequence of outcomes), multiply the probabilities along that path.

Using a probability tree:

• Labelling branches: Write the probability as a fraction on each branch. From any single node, branch probabilities must sum to 1.
• Multiplication rule: $P(A \text{ and } B) = P(A) \times P(B)$. For any path: multiply all branch probabilities along it.
• Adding paths: If the event can happen by multiple paths, add the probabilities of all those paths. E.g. P(exactly one red) = P(BR) + P(RB).

Probability tree and multiplication rule: P(A and B) = P(A) × P(B|A). For independent events P(B|A) = P(B), so P(A and B) = P(A) × P(B). Add probabilities of all branches that satisfy the required event.

Pause — copy P(A and B) = P(A) × P(B|A), the independence simplification P(A and B) = P(A) × P(B), and the addition rule for multiple paths: add path probabilities when the event can occur in more than one way into your book.