Is π Random?

Randomness has baffled human minds for ages. Now, it has become a more tractable problem for the human intellect. Thanks to probability theory.

But, we still have not been able to solve one problem correctly.

How to generate a totally random sequence of digits?

Human beings have tried to define pseudo-randomness and quantify what "we" want when we look for "totally" random digits The irony is that we have used deterministic algorithms to generate an indeterministic sequence of digits.

In 1927, a statistician by the name of L.H.C. Tippett produced a book titled Random Sampling Numbers. The contents of this book are 41,600 digits (from 0 to 9 ) arranged in sets of 4 in several columns and spread over 26 pages. It is said that the author took the figures of areas of parishes given in the British census returns, omitted the first two and last two digits in each figure of the area, and placed the truncated numbers one after the other in a somewhat mixed way till 41,600 digits were obtained. This book which is nothing but a haphazard collection of numbers became the best seller among technical books. A book of random numbers! A meaningless and haphazard collection of numbers, neither fact nor fiction.

C.R.Rao shares in his book "Statistics and Truth", that writing, when he was teaching the first-year class at the Indian Statistical Institute, he used to send my students to the Bon-Hooghly Hospital near the Institute in Calcutta to get a record of successive male and female births delivered. Writing M for male birth and F for female birth we get a binary sequence like the one obtained above by repeatedly tossing a coin or drawing beads.

In an article published in the International Statistical Review, Y. Dodge traces the 4000-year old history of π?and raises the question of whether the decimal digits of π?form a random sequence. Technically speaking,

A random sequence of symbols is a sequence that cannot be recorded by means of an algorithm in a form shorter than the sequence itself.

In this strict sense, the sequence of decimal digits in π?doesn't form a random sequence Ideally, the digits of π?will be random if no digit is preferred over another.

How to statistically verify this phenomenon?

Let's define the problem probabilistically, i.e. the random variable involved in the process.

Each such X is a discrete random variable that takes values in {0, 1, 2, 3, ..., 9}.

Let's take a sample of data - the first 100 digits of π. We assume that each digit is independent of the other digit.

Let's make a summary of the data in form of a frequency table.

Let's form the statistical testing procedure.

Test Statistic

From the given data, the observed value of the test statistic X = 4.2.

Under the null hypothesis, the test statistic follows the Chi-Square Distribution with 9 degrees of freedom.

p-value

We get the p-value to be 0.897763, suggesting strong evidence in favor of the Null Hypothesis.

I will end with the question to you, Is π Random?

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References:

• Statistics and Truth by C.R.Rao

Edits and Notes

• As Abhimanyu Gupta has pointed out in his comments with a beautiful example that equal proportions don't mean it is random. So, I must add that equal proportions are just one step in statistically showing the existence of randomness. Just with this one test, we cannot say π is random. We need more statistical tests. One such test is to count the number of runs of outcomes, and gaps, and match it with the expected outcomes. There are other randomness tests too. You can learn more about it here.
• As Chetan Srinivasa Kumar has rightly pointed out that the digits of π may not be independent. He is totally correct about it. Testing independence is another not so clearly defined problem in probability and statistics. But, one can use basic tests to reject non-independence ness, like autocorrelation tests. I am sharing the Autocorrelation Plot (ACF) of the first 100 digits of pi. Observe that it has no linear dependence in a certain lag.