Penney's Game, explained.
Penney’s game races coin-flip patterns until one appears first; the competition is nontransitive because overlap gives different patterns advantages against different rivals.
Why it happens
Every particular three-flip block has probability one eighth under a fair coin. But the game uses overlapping blocks in one continuous stream, so partial matches and interruptions affect the first-arrival race.
Equal probabilities of individual three-flip blocks do not imply equal chances of appearing first in an overlapping sequence.
Read the result
Choose a pattern, inspect the opponent’s counter-pattern and race several times. The outcome records the actual generated sequence. Wins in a handful of races need not reflect the exact long-run advantage.
A worked example
The HHH versus THH race
You select HHH and the opponent selects THH.
If the first three flips are HHH you win immediately; after any tails, two subsequent heads complete THH before HHH.
The opponent’s advantage comes from the overlap structure, despite equal isolated three-flip probabilities.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
For your pattern abc, the bot selects not-b, a, b. Fair seeded flips continue until one pattern occurs, with a 200-flip safety limit and an explicit unfinished result.
Where this idea is useful
A practical use
Ordering and overlap matter when comparing competing sequences, even if their standalone frequencies match.
A common misconception
“Equal pattern frequencies mean an even race.”
First arrival in overlapping sequences is a different event from appearing in one isolated block.
What this explanation leaves out
- Fair independent coin flips are assumed. A single race is not an estimate of the exact long-run probability.
Does the counter-pattern guarantee a win?
No. It gives a probability advantage under fair independent flips, not certainty in an individual race.
Where are you comparing event frequencies when the actual question concerns order or first arrival?
Further reading
Explore the original research or the teaching reference behind this experiment.