GCSE Maths Practice: probability-scale

Understanding Probability with Multiple Coin Flips

This question focuses on finding the probability that at least one tail appears when flipping two fair coins. Although this may seem simple at first glance, it introduces important ideas for GCSE Maths: sample spaces, complementary events, and counting favourable outcomes. These concepts form the foundation for more advanced probability topics such as conditional probability, binomial distributions, and multi-stage experiments.

When dealing with coins, each flip has two equally likely outcomes: heads (H) or tails (T). Because the coin is fair, each outcome has a probability of \(1/2\). When flipping the coin twice, the total number of possible outcomes increases, but each combined outcome is still equally likely. Listing all possible outcomes is a powerful strategy that helps ensure all cases are considered without repetition or omission.

Building the Sample Space

To solve this problem accurately, we construct the full sample space. Since each flip has two possible outcomes, two flips give \(2 \times 2 = 4\) total outcomes. These are:

• HH (heads then heads)
• HT (heads then tails)
• TH (tails then heads)
• TT (tails then tails)

Each of these outcomes is equally likely because the coin is fair and each flip acts independently of the previous one.

Finding Outcomes with At Least One Tail

The phrase "at least one" is a key signal in probability problems. It means one or more, which often includes several different cases. Here, outcomes that contain at least one tail are:

• HT
• TH
• TT

Three of the four possible outcomes meet this condition. Therefore, the probability is the number of favourable outcomes divided by the number of total outcomes.

Step-by-Step Method

1. Write the sample space: HH, HT, TH, TT.
2. Identify all outcomes containing at least one T.
3. Count them: 3 favourable outcomes.
4. Divide by the total number of outcomes: 4.
5. The probability becomes \(3/4\).

Worked Example 1

"What is the probability of getting exactly one tail?" Only HT and TH qualify, giving 2 favourable outcomes out of 4. So the probability is \(2/4 = 1/2\).

Worked Example 2

"What is the probability of getting no tails?" Only one outcome qualifies—HH. The probability is \(1/4\). This also illustrates the complementary method: P(at least one tail) = 1 − P(no tails).

Common Mistakes

• Forgetting to list all outcomes, leading to missed cases.
• Assuming "at least one" means exactly one.
• Incorrectly thinking the probability must be \(1/2\) because tails appears on half the flips.
• Not recognising independence between flips.

Real-Life Applications

Situations involving repeated independent events are common in real life. For example, quality checks where each product has a chance of being defective, medical testing where results follow probabilistic patterns, or computer simulations where binary outcomes are repeated many times. Understanding small examples like coin flips builds intuition for complex compound events.

FAQ

Q: Do order and sequence matter?
A: Yes—HT and TH are different outcomes because the first flip and second flip are separate events.

Q: What does "at least one" really mean?
A: It includes any scenario with one or more tails, meaning multiple cases must be counted.

Q: Is there another method besides listing outcomes?
A: Yes—use the complement: P(at least one tail) = 1 − P(no tails).

Study Tip

For "at least one" questions, the complement method is often faster: calculate the probability of getting none of the required outcome, then subtract from 1. This approach becomes very efficient in larger sample spaces.