Multi-Stage Experiments (VCE SSCE Mathematical Methods): Revision Notes

Multi-Stage Experiments

What are multi-stage experiments?

Many probability problems involve experiments that happen in multiple sequential steps, called multi-stage experiments. These are experiments that can be thought of as occurring in more than one stage.

Examples of multi-stage experiments:

• Tossing two coins (or tossing one coin twice)
• Rolling two dice (or rolling one die twice)
• Tossing three coins

In these experiments, we need to consider the possible outcomes at each stage separately, and then combine them to find all possible overall outcomes.

Organizing outcomes with tree diagrams

When working with multi-stage experiments, tree diagrams provide a systematic and visual way to list all possible outcomes. A tree diagram uses branches to represent the possible results at each stage of the experiment.

Two-stage experiment: Tossing two coins

Let's consider tossing two coins. We can think of this in two stages:

• Stage 1: The outcome from the first coin
• Stage 2: The outcome from the second coin

How to read a tree diagram:

• Each complete path from the start to the end represents one possible outcome
• We read along the branches from left to right
• The order of outcomes matters (HT is different from TH)

For two coin tosses, the sample space (the set of all possible outcomes) is:

$$\varepsilon = \{HH, HT, TH, TT\}$$

This gives us 4 equally likely outcomes.

Calculating probabilities in multi-stage experiments

When all outcomes in a multi-stage experiment are equally likely, we can calculate the probability of an event using this formula:

$$\text{Pr}(\text{event}) = \frac{\text{number of outcomes in the event}}{\text{number of outcomes in the sample space}}$$

Using tables for two-stage experiments

For some two-stage experiments, a table can be more convenient than a tree diagram. This works particularly well when both stages have several possible outcomes, such as rolling two dice.

Two dice experiment

When rolling two dice (or rolling one die twice), there are 6 possible outcomes on die 1 and 6 possible outcomes on die 2. We can display all possible combinations in a table:

The table shows all 36 equally likely outcomes as ordered pairs $(a, b)$, where:

• $a$ is the result on die 1 (the first die or first roll)
• $b$ is the result on die 2 (the second die or second roll)

Three-stage experiments

When an experiment involves more than two stages, tree diagrams become essential for organizing all possible outcomes. Tables don't work well for experiments with more than two stages.

Three coin tosses

Consider tossing a coin three times. The tree diagram extends to three stages:

The sample space for three coin tosses is:

$$\varepsilon = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$$

This gives us 8 equally likely outcomes.

Key techniques for multi-stage experiments

Choosing your method:

• Two stages: You can use either a tree diagram or a table
• Three or more stages: Always use a tree diagram

Reading tree diagrams:

1. Start at the left (the beginning of the experiment)
2. Follow each complete path to the right
3. Record the outcome by reading the labels on each branch
4. Make sure to maintain the correct order

Calculating probabilities:

1. List all outcomes in the sample space
2. Identify which outcomes satisfy the event condition
3. Count the favourable outcomes
4. Divide by the total number of outcomes (if equally likely)