Why a Coin Flip Has No Memory
Independent trials explained with coin flips: each toss keeps its own chance, and a streak does not tilt the next one.
A fair coin is the cleanest model of an independent trial. Independence means the next toss does not consult the earlier ones. Heads or tails on toss number eight has the same chance it had on toss number one, if the coin and the toss stay fair. That sentence is small. Most of the confusion about streaks in games of chance is a refusal to apply it.
What independence does, and does not, claim
Independence is a statement about the mechanism. It says the probability now does not change because of the recent results. It does not say the coin must be fair. A coin that lands heads 60 percent of the time can still be independent from toss to toss. Each toss is 60 percent heads, including the toss that follows four heads. Bias and dependence are different defects.
Dependence is what you get when the mechanism removes outcomes. Deal a card and do not replace it. The next card is drawn from a smaller deck, so the probability changed. That is real dependence. A roulette wheel that is spun again, or a pair of dice that is thrown again, is modeled as independent because the earlier result is not removed from the device. The companion essay on why streaks do not predict the next result follows that mistake into the casino. This one stays with the coin until the arithmetic is obvious.
Two tosses, four equally likely paths
A fair coin has two results, H and T, each with probability 1/2. Two independent tosses have four paths: HH, HT, TH, TT. Each path has probability 1/4, because 1/2 times 1/2 is 1/4. HT is not more likely than HH. The alternation looks more "mixed" to a human eye. The coin does not prefer a mixed look.
The chance of at least one head in two tosses is 3/4, because three of the four paths contain a head. The chance of two heads is 1/4, not 1/2. Adding the tosses does not add the probabilities. You multiply along a path and you add across distinct paths. Those two operations are the whole algebra of this example.
A streak does not load the next toss
Four heads in a row has probability 1/16 on a fair coin, since (1/2) to the fourth power is 1/16. Suppose you have just seen those four heads. The fifth toss is still 1/2. The 1/16 described a sequence that is now over. It is not a debt stored inside the coin. People who say the fifth toss is "due" for tails are applying the 1/16 backward, as if the coin had to balance a ledger by Friday.
Over a long series, about half the tosses will be heads if the coin is fair. That long-run share does not reach backward and edit the next toss. It is a description of a large pile of tosses, most of which have not happened. The next one happens alone.
Where the model would fail
Independence fails if the setup cheats. A person who glances at a coin and places it, a wheel that is not spun, a draw that is typed onto a screen by hand: these can make the next result depend on a choice, not on a fresh trial. The mathematical statement does not protect you from a crooked device. It tells you what a fair device would do.
You test the device by looking at the procedure, not by reading a short streak. Four heads are common enough, at 1 in 16 sequences of four, that they are terrible evidence of a trick. A coin that you never see tossed, combined with a payout that only the tosser calculates, is stronger evidence of a problem. Procedure first. Streaks are a late and weak clue.
Using the idea without a table
You can practice on paper. Write eight planned tosses as blanks. Fill them with a real coin. Before each toss, write the probability you assign to heads. If you understand independence, you write 1/2 every time, including after a run. If you notice yourself writing 1/3 because "tails is due," you have caught the habit the games rely on. No money needs to be involved. The lesson is the unchanged fraction.
Nothing here is a betting system, and a coin game is still subject to local law and age limits. The arithmetic holds in a kitchen.
Frequently asked questions
Does a fair coin remember a long run of heads?
No. If the tosses are independent, the next toss stays at 1/2 regardless of the run you just saw.
Is a biased coin the same problem as dependence?
No. A biased coin can still be independent. Each toss keeps the same unequal chance. Dependence means the chance itself changes because of earlier results.
Why is heads-then-tails as likely as two heads?
Because each toss contributes a factor of 1/2, and 1/2 times 1/2 is 1/4 for every specific two-toss path.