| Why does random imply all finite sequences will occur somewhere in the
infinite sequence? |
| Suppose we have an infinite sequence of 0s and 1s, and nowhere in the
sequence do we find 000. |
| By the definition given, this infinite sequence is not random. Why? |
| Suppose we're describing the sequence by listing all its terms. |
| Whenever we get to
a 00 pair, we don't have to say what the next number is. |
| It MUST be 1, because otherwise
the infinite sequence would contain 000. |
|
| Consequently, we don't have to list the entire
infinite sequence to specify it completely. We say only once that the sequence
does not contain the
triple 000, and then whenever the pair 00 occurs, we know the next number must be 1. |
| Similar arguments show that all finite sequences must occur somewhere
(in fact, infinitely often) in an infinite random sequence. If any one is missing, we
can use this missing sequence to describe the infinite sequence without listing all its
entries. |