Practice Maths

Two-Step Experiments and Tree Diagrams

Key Ideas

Key Terms

Multi-stage experiment
An experiment with two or more steps performed in sequence (e.g. flip a coin then roll a die).
Tree diagram
A diagram showing all possible outcomes by drawing branches for each stage of an experiment.
Branch
A line in a tree diagram representing one possible outcome at a stage, with the probability written on it.
Sample space
The complete set of all combined outcomes of a multi-stage experiment (listed at the ends of branches).
Combined outcome
The result of all stages together (e.g. HT when a coin is flipped twice).
With replacement
After each selection, the item is returned; probabilities remain the same for every stage.
Without replacement
After each selection, the item is not returned; the total and favourable outcomes change for subsequent stages.

Building a Tree Diagram

Start from a single point. Draw one branch for each possible outcome at Stage 1. From the end of each branch, draw further branches for each outcome at Stage 2.
Write the probability on each branch and the combined outcome at the end.

Hot Tip All branch probabilities at each stage must add to 1. Check this before multiplying. When reading outcomes from a tree diagram, work from left to right, multiplying as you go.

Worked Example

Question: A fair coin is flipped twice. Find P(one head and one tail in any order).

Step 1 — Draw the tree diagram.

  Flip 1    Flip 2    Outcome    P
  H ──── H     HH     1/2 × 1/2 = 1/4
    └──── T     HT     1/2 × 1/2 = 1/4
  T ──── H     TH     1/2 × 1/2 = 1/4
    └──── T     TT     1/2 × 1/2 = 1/4

Step 2 — Identify favourable outcomes: HT and TH.

Step 3 — Add their probabilities:
P(one head, one tail) = 1/4 + 1/4 = 1/2

What Is a Two-Step Experiment?

A two-step experiment (also called a two-stage experiment) is one where two actions or events happen in sequence. For example: flipping a coin and then rolling a die; drawing two cards from a deck one after another; spinning a spinner twice. Each "step" produces its own set of outcomes, and the combined result is a pair of outcomes — one from each step.

For a single event, listing the sample space is easy. But for a two-step experiment, there are many more combined outcomes to keep track of. A tree diagram is the best tool for this job because it organises the outcomes visually and ensures you do not miss any.

How to Draw a Tree Diagram

A tree diagram starts with the first step. Draw branches for each possible outcome of step 1. Then, from the end of each branch, draw a new set of branches for each possible outcome of step 2. Each complete path from the start to a final branch represents one combined outcome of the experiment.

Example: Flip a coin (H or T), then roll a die (1, 2, 3, 4, 5, 6).

  • First step: H or T (2 branches)
  • Second step: from each of H and T, draw 6 branches (for 1 to 6)
  • Total combined outcomes: 2 × 6 = 12 outcomes (H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6)

Write the outcome at the end of each final branch and the probability on each branch.

Multiplying Along Branches

To find the probability of a combined outcome, multiply the probabilities along the branches from start to finish. This is because the two events happen independently — the outcome of the first event does not change the probability of the second event.

Example (coin then die): P(H and 3) = P(H) × P(3) = 1/2 × 1/6 = 1/12.

This works whenever the two steps are independent — that is, the first step does not affect the probabilities of the second step. If the steps are not independent (e.g. drawing cards without replacement), the second-step probabilities change depending on what happened first — you must adjust the denominators on those branches.

Adding Across Branches for the Same Outcome

Sometimes the same event can happen via multiple different paths through the tree. To find the total probability of that event, add the probabilities of all paths that lead to it.

Example: Flip a coin twice. What is P(exactly one Head)?
Paths that give exactly one Head: {HT} with probability 1/2 × 1/2 = 1/4, and {TH} with probability 1/2 × 1/2 = 1/4.
P(exactly one Head) = 1/4 + 1/4 = 1/2.

The rule is: multiply along branches (for a single path), then add the results for different paths (for multiple paths leading to the same event).

Systematic Listing Without a Tree Diagram

For small experiments, you can also list all outcomes systematically in a grid or table. For example, rolling two dice can be shown in a 6 × 6 grid, giving 36 total outcomes. Then count the favourable outcomes and divide by 36.

A tree diagram is better when probabilities are unequal (e.g. a biased spinner), because you can write different probabilities on each branch. A grid is quicker when all outcomes are equally likely and there are only two steps.

Key tip: The two key rules for tree diagrams are: (1) multiply along branches to get the probability of one specific combined outcome, and (2) add across branches to get the total probability of an event that can happen in more than one way. A good check: all the final branch probabilities should add up to 1.

Mastery Practice

  1. List all outcomes in the sample space for each two-step experiment. Fluency

    1. A coin is flipped, then a four-sided die (1–4) is rolled.
    2. A standard die is rolled twice.
    3. One card is chosen from {A, B, C} and one from {1, 2, 3}.
    4. A spinner with sections Red and Blue is spun twice.
    5. A marble is drawn from {Red, Green, Blue} and replaced, then another is drawn.
    6. Two letters are chosen from {X, Y, Z} with replacement.

  2. Draw a tree diagram and find the probability of each event. Fluency

    1. A fair coin is flipped twice. Find P(two tails).
    2. A fair coin is flipped twice. Find P(at least one head).
    3. A fair coin is flipped three times. Find P(exactly two heads).
    4. A spinner has three equal sections: Red (R), Blue (B), Green (G). It is spun twice. Find P(same colour both times).

  3. Use the multiplication rule to find each probability without drawing a full tree diagram. Fluency

    1. A die is rolled and a coin is flipped. P(rolling a 6 AND getting tails).
    2. A bag has 3 red and 5 blue marbles. One marble is drawn and replaced, then another is drawn. P(red then blue).
    3. A bag has 4 green and 6 yellow marbles. With replacement: P(yellow then yellow).
    4. A spinner (1, 2, 3, 4 equal sections) is spun twice. P(sum equals 5).
    5. P(rolling an even number on a die) × P(flipping tails) — find this combined probability.

  4. Without replacement — draw a tree diagram or use reasoning to find the probability. Understanding

    1. A bag has 3 red and 2 blue marbles. Two marbles are drawn without replacement. Find P(both red).
    2. Same bag (3 red, 2 blue). Find P(one red and one blue) in any order.
    3. A box has 4 defective and 6 non-defective items. Two are selected without replacement. Find P(both non-defective).
    4. From a group of 5 students (3 boys, 2 girls), two are chosen at random without replacement. Find P(both girls).

  5. Analyse tree diagrams and combined outcomes. Understanding

    1. Two dice are rolled. Find P(both dice show the same number).
    2. Two dice are rolled. Find P(the sum is at least 10).
    3. A spinner has 4 equal sections (1, 2, 3, 4) and is spun twice. Find P(the product of the two results is even).
    4. A student randomly chooses one subject from {Maths, English} and one sport from {Swimming, Tennis, Athletics}. List all possible combinations and find P(Maths and Swimming).
    5. A bag has 2 red, 3 blue and 1 green marble. One marble is drawn (with replacement), then another. Find P(getting at least one red marble).

  6. Tree diagram problem solving. Problem Solving

    1. A game involves flipping a coin and rolling a die. You win if you get heads AND a number greater than 4. Find P(winning) and P(losing).
    2. At a school tuck shop, students choose one main (pie or sandwich) and one drink (water, juice, or milk). Draw a tree diagram showing all possible choices. How many outcomes are there? What is P(pie and juice) if all outcomes are equally likely?
    3. A bag has 5 red and 3 blue marbles. Two marbles are drawn without replacement.
      1. Draw a tree diagram, labelling all branches with probabilities.
      2. Find P(both same colour).
      3. Find P(at least one blue).
    4. Lily claims that in a two-step experiment with replacement, “the probability of any specific outcome is always the same no matter the order.” Is she correct? Give an example to support your answer.
  7. Colour bag — with and without replacement. A bag contains 4 red marbles and 2 blue marbles. Two marbles are drawn one after the other.
    1. If drawn with replacement, find P(two red marbles).
    2. If drawn without replacement, find P(two red marbles). Compare your answer to (i) and explain the difference.
    3. Without replacement: find P(exactly one blue marble).
    Problem Solving
  8. Board game spinner. A board game uses a spinner with four equal sections labelled 1, 2, 3, and 4. A player spins twice and adds the two results to find their total score.
    1. How many outcomes are in the full sample space?
    2. Find P(total score = 6).
    3. Find P(total score is odd).
    4. Find P(total score is greater than 6).
    Problem Solving
  9. Three children. Assume a child is equally likely to be born male (M) or female (F). A family has three children.
    1. Draw a tree diagram and list all 8 outcomes in the sample space.
    2. Find P(all three children are the same gender).
    3. Find P(at least two girls).
    4. Find P(the second child is a boy).
    Problem Solving
  10. Evaluate a claim. Sam draws two cards from a bag containing the cards {1, 2, 3, 4, 5} without replacement. Sam claims: “The probability that both cards are odd is the same whether I draw with or without replacement.”
    1. Calculate P(both odd) with replacement.
    2. Calculate P(both odd) without replacement.
    3. Is Sam correct? Explain why the two probabilities are or are not equal, and what feature of the sample (if any) would make them equal.
    Problem Solving
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