What are the chances that the 900 trillion and eighth digit of Pi will be a 5?

1 out of 10?

Perhaps, but what are the chances that the first digit is a 5? All the digits of pi are fixed and can be calculated and nothing is left to chance.

I wonder if the distribution of numbers is even across the 10 trillion known digits?

If not… if 5 shows up a statistically significant number of times fewer (or more) than the average, then you could make a better guess.

In general there are two ways of interpreting probability. One is that it is simply a measure of our uncertainty or ignorance. The other is that certain things are inherently probabilistic. The second interpretation is sometimes given for quantum mechanics. In the case of a digit of pi, we can use the first interpretation. If we had perfect knowledge, the probability would be 1 or 0. I believe that the digits of pi satisfy all the standard tests of randomness, so the best that we can do is to say that the probability is one out of 10.

By way of analogy, suppose I flip a coin and ask the probability of it being heads. In point of fact, the probability is either 1 or 0, but given lack of knowledge, the best that can be said is that the probability is about one half.

it is 3, so odds are zero

The statistical properties of the decimal digits of pi and numerous other irrational numbers, all of whose decimal digits go on without apparent pattern, have been studied. They appear to be truly random, in the sense that each digit appears approximately 1/10 of the time; each specific pair of adjacent digits appears 1/100 of the time, and so on. So without computing the 900,000,000,000,008th digit it’s reasonabe to declare that the digit being a 5 has a probability of 10%.

You could argue that it’s not actually random but pseudo-random, because ultimately the digits are completely determined by the underlying mathematics. Still, we can only observe behavior and measure distributions.

Thus any finite sequence of digits (let’s say a string of a million zeroes, or tomorrow’s winning lottery combo) not only can occur but must occur in the unending string of digits of any irrational number. Moreover it must occur an infinite number of times !

Computing all those digits is itself a non-trivial problem in applied math and computer science, and better algorithms are always being sought. There are many well-known infiinite series that converge to pi but they generally converge too slowly to be practical.

A breakthrough came a few years ago with the discovery of an algorithm to directly compute any specified digit of pi (like in your question)—except that it applies ony to hexadecimal digits, not decimal digits as you’d like.

If you were betting, 10% would clearly be the line. It may not be random, but it’s close enough.

@6rant6 I would be careful as it isn’t really a question of probability as the person taking the bet might already know the number.

If we are talking base 16, it is very easy to find out any particular digit. In base 10, it seems very random.

@PhiNotPi , That really is amazing. I wonder how that formula could ever have been derived.

@PhiNotPi yeah, that’s what I was talking about.