How often do 100 tosses give exactly 50 heads?

The fifty heads paradox

Someone tosses a coin 100 times and counts the heads. Then another 100 times, and again. In what share of these series will there be exactly 50 heads?

Why: the peak has many neighbours

Picture 101 baskets, from "0 heads" to "100 heads", and series landing in them. The "50" basket gets the most, but not by much: 49 and 51 come up almost as often (7.8% each against 8.0%), 48 and 52 a little less, and so on. All these neighbours together take the rest. The most likely result is only slightly better than a dozen close rivals, so on its own it is rare.

That is the plain-words explanation, with no formulas. The exact numbers and where exactly 8% comes from are below, in the strict layer.

Exactly half gets rarer, roughly half gets more certain

The more coins, the harder it is to hit exactly half: there are more possible results, and each one becomes rarer. But landing "roughly at half", between 40% and 60% heads, gets easier and easier. Move the number-of-coins slider.

Both facts are true at once, and there is no contradiction. The share of heads presses ever closer to half: that is the law of large numbers. But the number of heads spreads ever wider, by about √n either way, so hitting one particular number gets harder.

What the binomial distribution is

Take n identical tries. Each succeeds with the same chance p, and the tries do not affect one another. Count how many succeed. The binomial distribution says how likely each possible number of successes is, from 0 to n. A coin, a die, a visitor who buys or doesn't, a part that is sound or faulty — it is all the same.

It is easiest to see with four coins. There are 2 · 2 · 2 · 2 = 16 different sequences of heads (H) and tails (T), all equally likely. Group them by the number of heads:

The numbers 1, 4, 6, 4, 1 are a row of Pascal's triangle: the number of ways to choose which coins land heads. The middle wins because the most paths lead there. Yet even here "exactly half" happens in only 6 cases out of 16, that is 37.5%.

The question Pepys sent to Newton

In 1693 the English official and famous diarist Samuel Pepys wrote to Isaac Newton with a question about dice. Which bet has the best chance?

A few more traps

• Two and two is not the most commonA family has four children. If boys and girls are equally likely, "two and two" happens in 6 families out of 16 (37.5%), while "three and one" either way happens in 8 out of 16 (50%). The single most likely split loses to the pair of less even ones.
• Expecting one sixIn 6 rolls of a die one six comes up on average. But exactly one happens only 40.2% of the time, and none at all 33.5%. A third of attempts fall short of even the average.
• One in a thousand, a thousand triesIt feels like it should then happen "for sure". In fact it never happens 36.8% of the time. That is almost exactly 1/e, and this is how the binomial distribution for rare events turns into the Poisson distribution.
• A good free-throw shooterA player makes 70% of free throws. Out of 10 shots, 7 is the most likely number, but exactly 7 happens in only 26.7% of series. A coach drawing conclusions after 5 or 9 makes should remember that.

When to count successes the binomial way

• Buyers on a websiteEach of 1,000 visitors buys with the same chance. The number of purchases is binomial, so two identical pages give different numbers by pure chance. A/B tests are built on this.
• Defects in productionHow many faulty parts will a batch of 500 have if the defect rate is 1%? The binomial distribution tells you when 9 faulty parts is still ordinary chance and when it is a reason to stop the line.
• PollsEach person polled says "yes" with probability equal to the share of supporters. That gives the poll's margin of error: for 1,000 people and half supporters, ±3 percentage points.
• When it is not binomialIf the tries depend on one another (a sick person infects a neighbour, a buyer recommends to a friend), the chance changes from try to try, or you draw without replacement from a small urn, the formula gives wrong numbers. Usually the spread is then wider than it promises.

Why the peak falls like 0.8 / √n

From coins to the bell

In 1733 Abraham de Moivre was looking for a quick way to add up binomial probabilities for large n, since multiplying giant factorials by hand is impossible. He noticed that the bars of a binomial distribution with mean np and standard deviation √(np(1 − p)) lie on one and the same smooth curve. It was later named the normal distribution. That is why for large n all the slider charts above look like the same bell, only narrower.

Rare events: the Poisson limit

A little history

• Pascal and the triangleIn the 17th century Blaise Pascal tabulated the numbers of ways to choose k out of n. For a coin that is the binomial distribution itself, just without dividing by 2ⁿ.
• Jacob BernoulliHe derived the distribution for any chance p and proved the first law of large numbers. His book "Ars Conjectandi" came out in 1713, after the author's death. That is why a single such try is still called a Bernoulli trial.
• Newton calculated right, explained wrongIn 2006 the historian of statistics Stephen Stigler showed that Newton's numbers for Pepys's problem were flawless but his short verbal explanation was wrong: it would equally "prove" the answer for loaded dice, where the answer is different.