GCSE Maths Practice: listing-outcomes

Understanding Outcome Spaces for Multiple Coin Flips

This question explores a core idea in Higher GCSE probability: analysing all possible outcomes when several independent events occur in sequence. A coin has two possible results, Heads (H) or Tails (T). When flipping a fair coin three times, each flip is independent of the others, meaning the result of one flip does not affect the next. Because of this independence, the total number of possible outcomes can be calculated using exponentiation: two outcomes per flip, three flips, giving a total of 2³ = 8 different sequences.

Each outcome is written as an ordered sequence, for example (Heads, Tails, Heads). The order matters because (H, T, H) is different from (T, H, H). Higher GCSE probability places strong emphasis on recognising that sequential events create ordered outcomes. This specific question focuses not on calculating probabilities but on identifying which listed options represent valid sequences from the outcome space.

Step-by-Step Breakdown

1. Identify that each flip has two outcomes: H or T.
2. Recognise that independence allows multiplication of possibilities: 2 × 2 × 2 = 8.
3. List the full outcome set to understand all possibilities:

(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H), (T, H, T), (T, T, H), (T, T, T).

Any sequence made up only of H and T in a three-entry order is a possible outcome. All sequences in the question fit this rule.

Worked Example 1: Probability of Getting Exactly Two Heads

The favourable sequences are (H, H, T), (H, T, H), and (T, H, H). There are 3 such outcomes out of 8. Probability = 3/8.

Worked Example 2: Probability of Getting at Least One Tail

Only one outcome has no tails: (H, H, H). The remaining 7 include at least one T. So the probability is 7/8.

Worked Example 3: Probability of Getting Heads Every Time

Only one sequence — (H, H, H) — satisfies this. Therefore, the probability is 1/8.

Common Mistakes

• Forgetting that order matters. For example, (H, T, H) and (T, H, H) are different outcomes.
• Listing duplicates. Sometimes students confuse probability with counting and repeat sequences unnecessarily.
• Assuming more than two outcomes. A standard fair coin has exactly two outcomes unless stated otherwise.
• Mixing up sample space and favourable outcomes. The full sample space must be known before probabilities are calculated.

Why This Is a Higher-Level Question

Even though coin flips seem simple, the requirement to analyse ordered sets and recognise the structure of exponential outcome spaces builds skills that students later use for binomial probability, conditional probability, and probability trees. Higher-level reasoning involves understanding independence, sequential outcomes, and how sequences form a complete sample space.

Real-Life Applications

Sequential probability appears frequently: genetic patterns (dominant/recessive traits), computer algorithms, encryption, random number generation, and decision modelling. Understanding how repeated independent events combine is essential in data science, modelling, simulations, and statistics.

FAQ

Q: Do the flips influence each other?
A: No, coin flips are independent.

Q: Could a sequence like (Heads, Tails, Coin Stands on Edge) appear?
A: No. Only H and T are valid outcomes unless the question explicitly allows otherwise.

Q: Do we care about the order?
A: Yes. Each ordered sequence is a separate outcome in the sample space.

Study Tip

Whenever dealing with repeated independent events, write the sample space using a tree diagram or a sequence list. This helps avoid mistakes and strengthens your understanding of how probabilities combine.