Miracle Calculator (Littlewood’s law: how often do one-in-a-million events happen?)

Understanding the Littlewood's law miracle calculator — concept and principle

This calculator applies Littlewood's law to show how often a "one-in-a-million" event should happen to an ordinary person, given how many events they notice each day. It is a fun way to understand coincidences and why surprising things happen more often than intuition suggests.

Rare events, many chances: with the default inputs the result is 34.7 days (expected wait for one).

How it is calculated

Events per day n = events per second × alert hours × 3,600.
Expected wait for one event = N ÷ n days, for odds of 1 in N.
Expected number in D days = n × D ÷ N.
Chance of at least one per day = 1 − (1 − 1/N)^n; in the period = 1 − (1 − 1/N)^(n × D).
Days until a 50% chance = ln 0.5 ÷ (n × ln(1 − 1/N)).

The principle

J. E. Littlewood observed that if you are awake and alert for about 8 hours a day and notice roughly one "event" per second, you experience about a million events in about 35 days. So an event with odds of a million to one should happen to you about once a month. Each event is treated as an independent chance, and with very many chances the probability of at least one rare event becomes high. The same logic explains why, in a large population, some people win lotteries twice or meet a long-lost friend abroad.

Worked example

Default inputs: odds 1 in 1,000,000, 1 event per second, 8 hours a day, 365 days.

Events per day = 1 × 8 × 3,600 = 28,800.
Expected wait = 1,000,000 ÷ 28,800 = 34.7 days.
Expected number in a year = 28,800 × 365 ÷ 1,000,000 = 10.5.
Per day: 1 − (1 − 10⁻⁶)^28,800 = 2.84%; in the year this is effectively 100%.
50% chance after ln 0.5 ÷ (28,800 × −0.000001) = 24.1 days.

Each point is the calculator re-run with only “Events you notice per second” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Events you notice per second	Expected wait for one such event (days)
0.25	138.9
0.5	69.4
0.75	46.3
1	34.7
1.25	27.8
1.5	23.1
2	17.4

How to read the result

The point is not to predict miracles but to show that rare events are common when there are many opportunities. When something astonishing happens, it helps to ask how many other coincidences could have happened and would have seemed just as remarkable. That is why statisticians are cautious about treating a single surprising event as evidence of something special.

Common mistakes

Treating the result as a forecast of a specific event.
Ignoring how many different coincidences would count as "a miracle".
Assuming events are independent when they are linked.
Reading 100% as absolute certainty; it is rounded.

Keep in mind: This is an illustration of probability, not a prediction. The inputs are rough assumptions about attention and events.

Frequently asked questions

What is Littlewood's law?

The idea that a person experiences about one "one-in-a-million" event per month, given about one event per second for 8 hours a day.

Why do coincidences happen so often?

Because there are so many opportunities each day that some rare combinations become likely.

How long until a one-in-a-million event is 50% likely?

About 24 days with the default assumptions.

Sources: J. E. Littlewood, Littlewood's Miscellany (Cambridge University Press, 1986)

Checked 9 Nov 2026. Littlewood’s law as stated in J. E. Littlewood, “Littlewood’s Miscellany” (1986), summarised by Improbable Research (2013): one event per second, 8 alert hours a day, one-in-a-million events about every 35 days. Probability of at least one success in independent trials.

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