did you know a coin has no memory of its last flip?

if flips are independent and the coin is fair, the next flip is still fifty-fifty after five heads. randomness does not keep a little ledger and repair yesterday's imbalance.

our eyes are good at spotting patterns. but any particular sequence of six flips is just as unlikely as six heads: HHTHTT and HHHHHH each describe exactly one of 64 possible sequences.

before flipping, six heads has probability 1/64. after five heads have already happened, only one uncertain flip remains. its chance of heads is 1/2. the earlier outcomes are now known, not future hurdles.

over many flips, the proportion of heads tends towards half. that does not require the absolute difference between head and tail counts to shrink, or tails to become more likely.

knowing the previous result does not change the next result's probability. multiplying one-half six times describes six specified flips before any have happened.

P(H next | HHHHH already) = 1/2; P(HHHHHH beforehand) = (1/2)⁶

• a long streak can be evidence about an unknown coin's bias. that changes your knowledge of the coin, not the coin's memory.
• independence is an assumption about the mechanism. real experiments can break it if the launching method systematically links outcomes.