Miracle Calculator

What Is Littlewood's Law of Miracles?

Littlewood's Law of Miracles was formulated by the British mathematician John Edensor Littlewood in his 1953 publication A Mathematician's Miscellany. It states that an individual can statistically expect to experience an event with a one-in-a-million probability (defined mathematically as a "miracle") approximately once every 35 days.

Rather than invoking supernatural intervention, Littlewood's law demonstrates the Law of Truly Large Numbers: given a sufficiently large number of opportunities or observations, highly improbable events are not only possible but virtually guaranteed to occur regularly. You can also analyze event distributions with our Statistics Calculator and Sample Size Calculator.

The Mathematical Formulation

Littlewood based his derivation on several straightforward assumptions about human perception and wakefulness:

1. An alert human is awake and perceives events for approximately 8 hours per day.
2. A human consciously experiences roughly 1 event per second (such as seeing a face, hearing a sound, thinking a thought, or interacting with an object).
3. A "miracle" is defined quantitatively as an event having a probability of \(p = 10^{-6}\) (one in a million).

Under these conditions, we can calculate the total number of events experienced per day:

$$\text{Events per day} = 8 \text{ hours} \times 60 \text{ minutes} \times 60 \text{ seconds} = 28,800 \text{ events/day}$$

To accumulate \(1,000,000\) events, the required number of days is:

$$\text{Days per miracle} = \frac{1,000,000}{28,800} \approx 34.72 \text{ days} \approx 35 \text{ days}$$

Expected Miracles Over Time

For any arbitrary duration of \(D\) days with \(H\) awake hours per day and an observation rate of \(r\) events per second, the total sample of events \(N\) is:

$$N = D \times H \times 3600 \times r$$

The expected number of miracles \(\mu\) (the mean of the binomial or Poisson distribution) is given by:

$$\mu = N \times p = \frac{N}{\text{Odds}}$$

Poisson Probability of Experiencing Miracles

Because \(N\) is very large and \(p\) is very small, the distribution of miracles closely follows a Poisson distribution with parameter \(\lambda = \mu\). The probability of experiencing exactly \(k\) miracles in that period is:

$$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$

The probability of experiencing at least one miracle is the complement of experiencing zero miracles:

$$P(X \ge 1) = 1 - P(X = 0) = 1 - e^{-\lambda}$$

Over a period of 35 days where \(\lambda \approx 1\), the chance of witnessing at least one miracle is:

$$P(X \ge 1) = 1 - e^{-1} \approx 1 - 0.3679 = 63.21\%$$