Coin flipper

What to expect from n flips

The number of heads in n flips of a fair coin follows a binomial distribution with p = ½. Its average and spread are:

expected heads = n ÷ 2 SD = √(n × ½ × ½) = √n ÷ 2

The calculator reports how far your count landed from n ÷ 2 in standard deviations (a z-score), and the exact probability that a fair coin gives a split at least that far from even, in either direction. That probability is the two-sided p-value of a binomial test for a fair coin.

A worked example

The default flips 10 coins, so you should expect 5 heads with a standard deviation of √10 ÷ 2 = 1.58. Suppose a toss comes up with 8 heads and 2 tails.

z = (8 − 5) ÷ 1.58 = 1.90

A split of 8–2 or more extreme means 8, 9 or 10 heads, or 8, 9 or 10 tails. Counting the sequences: C(10, 8) + C(10, 9) + C(10, 10) = 45 + 10 + 1 = 56 on each side, out of 2¹⁰ = 1,024.

P(split at least as uneven as 8–2) = 2 × 56 ÷ 1024 = 0.109

So about one toss of 10 coins in nine is at least this lopsided. That is ordinary luck, not evidence of a trick coin. Press Flip a few times and you will see splits like this turn up.

How to read the result

Small numbers of flips swing a long way. With 10 coins, anything from 2 to 8 heads is well within normal variation. The percentage of heads settles as the number of flips grows, because the standard deviation of the proportion is 0.5 ÷ √n: about 16 percentage points for 10 flips, 5 points for 100 and half a point for 10,000. The raw count, on the other hand, wanders further from n ÷ 2 as n grows, since its standard deviation is √n ÷ 2.

The longest run is often longer than people guess. In 100 fair flips the longest run of the same face is usually 6 or 7, and a run of 5 heads somewhere in the sequence is more likely than not. The coin toss streak calculator gives the exact figures.

Common mistakes

• Expecting perfect alternation. Real random sequences have more long runs than hand-made “random” ones. A sequence with no run longer than 3 in 100 flips would be suspicious.
• Treating a 7–3 split as proof of bias. It needs many more flips before a small bias stands out from chance.
• Thinking tails is due after a run of heads. Each flip is independent of the last.

To work out exact probabilities rather than flip, use the coin flip probability calculator, which gives the chance of exactly, at least or at most k heads.