The Argument With a Broken Step
The motion: “After five heads in a row, tails is more likely on the next throw.”
Your opponent: Alan Turing · Course: Pearson Edexcel International GCSE Mathematics (4MA1) · Assessment objectives: AO1, AO3
Turing will argue it with complete confidence and a step that does not follow. Find it.
What you are asked to argue
Defeat the motion, justify the correct account of independence, then locate the invalid step in the argument put to you.
What the specification expects you to bring
- Independent events and what independence means formally
- The probability of a sequence versus the probability of the next event
- Relative frequency and how it behaves over many trials
- Tree diagrams and the multiplication of independent probabilities
Positions that are defensible
Disagreeing with the figure is not an error. Any of these can score full marks if argued well:
- That a real coin might be biased, so five heads is weak evidence FOR heads — a genuinely strong answer that inverts the fallacy
- That the motion is true of a sampling-without-replacement model, and the error is in the model, not the arithmetic
Common misconceptions this debate is built to expose
- Wrong: A run of one outcome makes the other more likely next time.
The correction: The coin has no memory. Each throw is independent, so the probability remains one half regardless of what preceded it. - Wrong: Six heads in a row is very unlikely, so the sixth head is unlikely.
The correction: The sequence is unlikely BEFORE any throw. Once five heads have occurred, only the sixth throw remains, and that is one half. - Wrong: The law of large numbers means the counts even out.
The correction: The PROPORTION converges to a half. The absolute difference between heads and tails does not shrink and typically grows.
What passing it proves
Independence stated formally, the sequence/next-event distinction made precisely, and the invalid step located and explained.
Why them?
Read the life before you argue with it — the case you are about to meet was built by a real person, over a real career:
- The life of Alan Turing — Father of Computer Science · Mathematician & Computer Scientist · 1912–1954
Prepare for this debate → — the motion, the rounds and the marking descriptors, then the debate itself. Free, no account needed.