GCSE Maths Practice: listing-outcomes

Question 2 of 10

Learn how to count the total outcome space for multiple independent events, an essential skill for Higher GCSE probability.

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\( \begin{array}{l}\textbf{How many outcomes are possible when} \\ \textbf{flipping three fair coins?}\end{array} \)

Choose one option:

Map the outcomes using a tree diagram to avoid missing combinations.

Understanding Outcome Spaces in GCSE Probability

When dealing with probability, one of the most important ideas to master is the concept of an outcome space. The outcome space represents all the possible results that can occur in a situation. In GCSE Maths, especially at Higher level, questions often involve multiple independent events such as coin flips, dice rolls, or selections from a bag. Understanding how to list and count outcomes is essential for building more complex probability calculations later.

Independent Events

An event is independent when the result of one action does not affect the result of another. A coin flip is a classic example: whether the first coin lands on heads has no impact on the second or third. Each coin always has two possible outcomes: heads (H) or tails (T). When combining independent events, we multiply the number of outcomes for each event to find the total number of possible results.

Step-by-Step Method

  1. Identify the number of outcomes for a single event. For a coin, this is 2.
  2. Recognise that each flip is independent of the others.
  3. Multiply the outcomes together: for three flips, calculate \(2 \times 2 \times 2\).
  4. List outcomes systematically if needed, such as HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Worked Example 1

If you flip a coin twice, each flip has 2 outcomes. The total number of outcomes is \(2^2 = 4\). Possible outcomes include HH, HT, TH, and TT. This technique uses the same logic as the main question but applied to two events.

Worked Example 2

A bag contains two coloured counters: red and blue. If you randomly select one counter and then flip a coin, each action has a number of outcomes: selecting a counter has 2 outcomes, flipping a coin has 2 outcomes, so there are \(2 \times 2 = 4\) outcomes. Listing them helps: (Red,H), (Red,T), (Blue,H), (Blue,T).

Worked Example 3

A dice roll has 6 outcomes. Rolling a dice and flipping a coin produces \(6 \times 2 = 12\) possible results. Understanding this multiplication rule is crucial for probability trees later in the GCSE course.

Common Mistakes

  • Thinking three coins give 6 outcomes because there are three objects. This is incorrect: the number of outcomes is based on combinations, not quantity.
  • Listing outcomes without a consistent pattern, which can lead to missed or duplicated results.
  • Adding outcomes (2 + 2 + 2) instead of multiplying (2 × 2 × 2).

Real-Life Applications

This type of probability is used in computing, genetics, encryption systems, gaming, and simulations. Independent events underpin random number generators, decision trees, and modelling tools used in finance and science.

FAQ

Q: Why do we multiply outcomes?
A: Because each independent event expands the outcome space. Multiplication counts all combinations.

Q: Do orderings matter?
A: Yes. HTH and THH are different sequences.

Q: What if coins are biased?
A: The number of outcomes stays the same; only their probabilities change.

Study Tip

Whenever dealing with multiple independent events, write a small table or outcome tree. This keeps combinations organised and avoids errors in counting.

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