The odds of a coin flip streak, worked out

Two different questions get confused constantly with coin flips. "What's the chance of heads on the next flip?" is always 50%, forever, no matter what happened before. "What's the chance of getting heads five times in a row?" is a completely different number, and it shrinks fast. Both are simple arithmetic once you keep them separate.

The method

1. A single fair flip has exactly two equally likely outcomes, so P(heads) = P(tails) = 1/2. This never changes — a coin has no memory of previous flips, so "it's due for tails" is not a real thing.
2. Independent events multiply. The chance of two particular outcomes in a row (say, heads then heads) is 1/2 × 1/2 = 1/4, not 1/2 + 1/2.
3. For a streak of n flips all landing the same way, the probability is (1/2)^n.
4. Flip the fraction over and you get "1 in how many": 1 / (1/2)^n = 2^n. A streak of n is a 1-in-2^n event.
5. This is exactly the same math as a PIN or password keyspace — each additional flip (or digit, or character) multiplies the space by the same factor, here 2 instead of 10.

A worked example

Using this site's coin flip tool, each click is one independent 50/50 draw. Here is what a streak of identical results actually costs in probability:

• 2 in a row: (1/2)^2 = 1/4 = 0.25, or 1 in 4.
• 3 in a row: (1/2)^3 = 1/8 = 0.125, or 1 in 8.
• 5 in a row: (1/2)^5 = 1/32 ≈ 0.03125, or 1 in 32.
• 10 in a row: (1/2)^10 = 1/1024 ≈ 0.000977, or 1 in 1,024.

Ten heads in a row looks impossible when you watch it happen, but across enough flips it is routine: flip a coin 1,024 times and a 10-heads streak showing up somewhere in that run is a near-coin-flip itself, because there are hundreds of overlapping 10-flip windows to check, not just one.

Why "it's due" is wrong

After four heads in a row, the fifth flip is still exactly 50/50 — the coin does not track its own history. The 1-in-32 figure for "five heads in a row" describes the whole sequence viewed in advance, before any flip happened. Once the first four are already heads, you're no longer asking about a five-flip sequence; you're asking about one flip, and that one flip is 1/2 like every other. Mixing up those two questions is the entire gambler's fallacy in one sentence.