Flip a Coin

One bit, two outcomes

A coin flip is a Bernoulli trial: P(heads) = P(tails) = 1/2. The tool asks the browser for one random bit and maps the two possible values to Heads and Tails. That model assumes neither side has been given a weight and that the choice is made before the result is seen.

A decision with an exact split

Suppose two roommates both want the last parking space and agree that either person may have it. Assign Maya to Heads and Jordan to Tails before clicking. If Heads appears, Maya gets the space; if Tails appears, Jordan does. Each person has probability 0.5, so over 200 independent disputes the expected count for either person is 200 × 0.5 = 100, although no short run must be exactly even.

What fairness does not settle

Equal odds do not make unequal options equally good. A coin cannot compensate for one person having already paid, a safety consequence, a rule in a lease, or a choice with more than two valid outcomes. Put those facts on the table before using chance; otherwise the random result merely hides a disagreement about the rule.

Independence is the important assumption

A fresh flip has no memory of the previous display. Four Heads in a row has probability (1/2)^4 = 1/16, but the next flip is still one-half Heads. Waiting for Tails because Heads has appeared repeatedly is the gambler's fallacy, not a balancing mechanism.

A useful record

For a consequential tie-break, write the two names, the assignment, and the time before the flip. That small procedure prevents the common error of changing what Heads means after the result appears. A single flip is suitable for a transparent two-way tie, not for allocating money, access, or duties that require an agreed policy.

A probability check

If you use a coin for a one-off tie, do not judge fairness from one result. For a test of 64 independent flips, a rough expectation is 32 Heads, yet values around that count are normal. The agreed mapping of face to person is the part that makes this particular decision auditable. A physical coin can also be a poor substitute where its landing surface, catching method, or visibility is disputed. This browser version has no claim to certify a real-world event; its value is that the two possible labels and the outcome are plainly displayed for a simple pre-agreed decision.

When a binary rule is actually fair

A coin is appropriate only after the two claims have been made comparable. For instance, it can settle which equally qualified speaker goes first, but it cannot decide whether someone should lose a reserved accessible space. The random bit is symmetrical; the surrounding rule must be symmetrical too. Naming the consequence before the flip makes that distinction visible.

Two outcomes make a binary commitment

A coin represents the sample space {heads, tails}. Under the fair-coin model, each outcome has probability 1/2, so neither side is evidence for the better option. That is precisely why it can settle a deadlock between two acceptable alternatives. For example, two teammates may agree that either Tuesday or Thursday works, assign Tuesday to heads and Thursday to tails, then accept the displayed outcome. The calculation is short, but the prior agreement matters: deciding which day heads means after the result turns a 50–50 experiment into a negotiation.

What ten flips do and do not show

With ten independent fair flips, the number of heads follows a binomial model. The expected number is 10 × 1/2 = 5, but five heads is not guaranteed. Exactly five heads can occur in 252 of the 1,024 equally likely ordered sequences, a probability of about 24.6%. Getting seven heads is not proof that the next flip should be tails; the next flip still has a 1/2 model probability. A streak is surprising only when judged before it happens, not after someone has selected it from a long list of possible patterns.

Physical fairness and procedural fairness differ

This browser result is a neutral two-way choice, not a measurement of a particular metal coin. A physical toss can be affected by its starting face, surface, launch, or catching method; a browser generator instead relies on its programmed random source. In either case, procedural fairness needs visibility. State the mapping aloud, perform one result, and do not retry because the answer is unpopular. If the decision involves money, safety, consent, or a third person who cannot agree to the rule, a coin is too thin a process: record criteria and choose on those criteria rather than outsourcing responsibility to one bit.

Use one flip, not an approval poll

A flip resolves exactly one binary question. It should not be used repeatedly until it produces permission for a decision someone already wants. If heads means take the train and tails means drive, one result completes the protocol. Asking for three flips and accepting the majority is another rule with different probabilities: a majority of heads in three fair flips occurs 4/8 of the time, still one half, but it takes longer and creates more room to abandon an unwanted result. Simplicity is a safeguard here.

Why a retry rule changes the odds

A stated retry rule is a different experiment, not a harmless way to make a flip feel safer. Suppose Heads means that a team presents first and the team may flip again once if Tails appears. The chance that Heads eventually appears is the chance of Heads on the first flip plus the chance of Tails followed by Heads: 1/2 + (1/2)(1/2) = 3/4. The apparently neutral rule now favors the label assigned to Heads. Repeating until a preferred face appears makes its eventual probability 1, which is selection rather than a tie-break.

Questions about this selection

Other chance methods

All coins and lots tools · Random Card Draw · Flip Multiple Coins