Flip Multiple Coins

Counting independent trials

With n fair coins, the number of Heads follows a binomial distribution: P(H = k) = C(n,k)(1/2)^n. Every coin is treated as an independent trial, so a result of seven Heads and three Tails from ten coins is a count, not a prediction that the next batch will look similar.

Ten flips with real probabilities

For the displayed batch of 10 coins, the chance of exactly 6 Heads is C(10,6)/2^10 = 210/1024, about 20.5%. The chance of at least 6 Heads is (210 + 120 + 45 + 10 + 1)/1024 = 386/1024, about 37.7%. The expected number of Heads is 10 × 1/2 = 5, but five is the average across many batches, not a required result for this click.

Why totals can look lopsided

All ten Heads has probability 1/1024, so it is unusual but entirely compatible with fair coins. A run is evidence of a problem only when there is an independently specified test and enough observations; eyeballing one surprising batch gives no reliable verdict about the generator.

Choose the count before sampling

This page models a fixed number of trials. Deciding to add flips only after seeing an inconvenient count changes the experiment and makes any percentage hard to interpret. For a classroom demonstration, state 'ten flips per group' first, then compare the groups after each has completed ten.

Where the binomial model ends

The formula does not apply to weighted coins, a deck of cards, or draws removed from a container. Those cases change either the probability of each outcome or the independence between draws. Use this tool for independent yes/no events and use a without-replacement draw when an item cannot appear twice.

A count check

Add the two displayed counts before interpreting the pattern. With ten requested coins, Heads plus Tails must equal ten; a total of nine or eleven is an input or display error, not an improbable outcome. This arithmetic check is more informative than expecting an exact five-to-five split. For a class experiment, collect several batches in a table before comparing the observed proportion with one-half. A total of 200 flips is more informative than one dramatic batch of ten, but it still measures random variation rather than proving that a particular coin or browser is perfect.

What the histogram is for

Repeated batches are useful when the question is about a distribution rather than a winner. Put each count of Heads in its own row: ten flips may yield 3, then 7, then 4. The average of many such rows tends toward five, while the individual rows remain variable. That is a different use from repeatedly flipping until a preferred side appears.

Count distributions rather than memorable strings

For n fair coins, the chance of exactly k heads is C(n,k)/2n. The coefficient C(n,k) counts the different orders that produce the same count. With five coins, exactly three heads has probability C(5,3)/32 = 10/32 = 31.25%, because HHTTT is only one of ten orders with three heads. By contrast, the exact ordered pattern HHTTT has probability 1/32. The distinction prevents a common reporting error: a count such as “three heads” combines many sequences, while a named sequence does not.

A six-coin classroom example

Flip six coins to model six independent yes-or-no responses. There are 64 possible ordered outcomes. The probability of no heads is 1/64, exactly one head is 6/64, exactly two heads is 15/64, exactly three heads is 20/64, and the remaining counts mirror those values. Thus three heads is the single most likely count at 31.25%, yet it occurs less often than “three or more heads,” whose probability is (20 + 15 + 6 + 1)/64 = 42/64 = 65.625%. Choose the event before flipping; changing from an exact count to a broad range after seeing the result changes the question.

Independent is an assumption about the experiment

Multiple virtual flips are modeled as separate trials with the same two probabilities. That model works for simulation, simple games, and a pre-agreed test. It does not turn a handful of outcomes into evidence about a real manufacturing process, a survey, or a player’s skill. A run of four tails in six flips can occur under a fair model: it needs no correction and does not make heads “due.” When the purpose is to assign people rather than study counts, a shuffled list is clearer because it can prevent repeats; a batch of coin flips deliberately allows any number of heads from zero to all n.

Plan for a threshold before pressing flip

Suppose a party game awards a bonus when at least four of five coins are heads. The chance is [C(5,4)+C(5,5)]/32 = 6/32 = 18.75%. That is very different from asking for “a lot of heads” after seeing the display. State whether a threshold is inclusive, whether all coins are visible together, and whether a failed attempt may be retried. Those details define the event; the random output only tells you whether that already-defined event occurred.

Use a range when the question is about variation

For ten fair coins, a count from three through seven Heads covers the central part of the distribution. Its probability is (120 + 210 + 252 + 210 + 120) / 1024 = 912/1024, or about 89.1%. That leaves roughly 10.9% for zero through two or eight through ten Heads combined. This calculation gives a classroom a concrete expectation without pretending that a 2–8 result proves anything is wrong. The interval was chosen before the batch; it is not a label attached afterward to whichever outcome looks ordinary.

Questions about this selection

Other chance methods

All coins and lots tools · Flip a Coin · Heads or Tails Best of Series