Five heads. Surely tails is next?
For a fair coin with independent flips, the probability of heads on the next flip is 1/2, or 50%—even after five heads. Fair means heads and tails have equal chances. Independent means earlier outcomes do not change the next flip’s chances.
Imagine choosing who washes the picnic mugs by flipping a coin. Five heads arrive in a row. Someone announces, “Tails is due!” The coin has received no such instruction. Cambridge’s NRICH explains why past flips do not control the next one.
Then why does the run feel surprising?
Before five flips, the probability of getting heads on all five is 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32, or 3.125%. That asks about an entire sequence that has not happened yet.
After five heads have happened, the next flip is a new question. Under our fair, independent model, its chance of heads is still 50%. A rare past run does not turn the next toss into a correction service.
Two flips make the difference visible
Write out the ordered possibilities for two flips: HH, HT, TH and TT. In this model, each has probability 1/4.
Exactly one head can happen in two ways: HT or TH. Its probability is therefore 2/4 = 1/2. But “heads, then tails” names only HT: 1/4. OpenStax works through these distinct ordered outcomes. A tiny change in wording changes what you count.
The small print worth remembering
The belief that tails becomes overdue is called the gambler’s fallacy. It mistakes random streaks for a rule requiring the next result to balance them.
Our answer assumes a fair coin and independent flips; it is not a promise that every physical coin or tossing method is perfectly balanced. As in working out what must be true, start with the assumptions. Then ask: am I predicting one new result, or a whole sequence?