# Random data

`# #![allow(unused_variables)] #fn main() { # extern crate rand; # use rand::RngCore; // get some random data: let mut data = [0u8; 32]; rand::thread_rng().fill_bytes(&mut data); #}`

## What is randomness?

What does **random** mean? Colloquially the word can mean simply *unexpected*
or *unknown*, but we need to be a bit more precise than that. Wikipedia gives us
a more specific definition:

Randomnessis the lack of pattern or predictability in events.

We can take this further: *lack of pattern* implies there is no *bias*; in
other words, all possible values are equally likely.

To understand what a *random value* is, we still need a context: what pool of
numbers can our random value come from?

- To give a simple example, consider dice: they have values 1, 2, 3, 4, 5 and 6, and an unbiased (fair) die will make each number equally likely, with probability ⅙th.
- Now lets take a silly example: the natural numbers (1, 2, 3, etc.). These
numbers have no limit. So if you were to ask for an unbiased random
natural number, 1, 5, 1000, 1 million, 1 trillion — all would be equally
likely. In fact, for
*any*natural number`k`

, the numbers`1, 2, ..., k`

are an infinitely small fraction of all the natural numbers, which means the chance of picking a unbiased number from this range is effectively`1/∞ = 0`

. Put another way: for*any*natural number, we expect an unbiased random value to be bigger. This is impossible, so there cannot be any such thing as an unbiased random natural number. - Another example: real numbers between 0 and 1. Real numbers include all the
fractions, irrational numbers like π and √2, and all multiples of those...
there are infinitely many possibilities, even in a small range like
`(0, 1)`

, so simply saying "all possibilities are equally likely" is not enough. Instead we interpret*lack of pattern*in a different way: every interval of equal size is equally likely; for example we could subdivide the interval`0,1`

into`0,½`

and`½,1`

and toss a coin to decide which interval our random sample comes from. Say we pick`½,1`

we can then toss another coin to decide between`½,¾`

and`¾,1`

, restricting our random value to an interval of size`¼`

. We can repeat this as many times as necessary to pick a random value between`0`

and`1`

with as much precision as we want — although we should realise that we are not choosing an*exact*value but rather just a small interval.

What we have defined (or failed to define) above are uniform random number
distributions, or simply **uniform distributions**. There are also non-uniform
distributions, as we shall see later. It's also worth noting here that a
uniform distribution does not imply that its samples will be *evenly* spread
(try rolling six dice: you probably won't get 1, 2, 3, 4, 5, 6).

To bring us back to computing, we can now define what a uniformly distributed random value (an unbiased random value) is in several contexts:

`u32`

: a random number between 0 and`u32::MAX`

where each value is equally likely`BigInt`

: since this type has no upper bound, we cannot produce an unbiased random value (it would be infinitely large, and use infinite amounts of memory)`f64`

: we treat this as an approximation of the real numbers, and,*by convention*, restrict to the range 0 to 1 (if not otherwise specified). We will come back to the conversions used later; for now note that these produce 52-53 bits of precision (depending on which conversion is used, output will be in steps of`ε`

or`ε/2`

, where`1+ε`

is the smallest representable value greater than`1`

).

## Random data

As seen above, the term "random number" is meaningless without context. "Random
data" typically means a sequence of random *bytes*, where for each byte, each of
the 256 possible values are equally likely.

`RngCore::fill_bytes`

produces exactly this: a sequence of random bytes.

If a sequence of unbiased random bytes of the correct length is instead
interpreted as an integer — say a `u32`

or `u64`

— the result is an unbiased
integer. Since this conversion is trivial, `RngCore::next_u32`

and
`RngCore::next_u64`

are part of the same trait. (In fact the conversion is
often the other way around — algorithmic generators usually work with integers
internally, which are then converted to whichever form of random data is
required.)