One-digit numbers:
It can be any of these 8 digits: 1,2,3,4,5,6,7,9
So there are 8 one-digit numbers without any 8's.

Two-digit numbers:
The first digit can be any of these 8 digits: 1,2,3,4,5,6,7,9
The second digit can be any of these 9 digits: 0,1,2,3,4,5,6,7,9
So there are 8×9 two-digit numbers without any 8's.

Three-digit numbers:
The first digit can be any of these 8 digits: 1,2,3,4,5,6,7,9
The second digit can be any of these 9 digits: 0,1,2,3,4,5,6,7,9
The third digit can be any of these 9 digits: 0,1,2,3,4,5,6,7,9
So there are 8×9×9 three-digit numbers without any 8's.

Four-digit numbers:
The first digit can be any of these 8 digits: 1,2,3,4,5,6,7,9
The second digit can be any of these 9 digits: 0,1,2,3,4,5,6,7,9
The third digit can be any of these 9 digits: 0,1,2,3,4,5,6,7,9
The fourth digit can be any of these 9 digits: 0,1,2,3,4,5,6,7,9
So there are 8×9×9×9 three-digit numbers without any 8's. 

Five-digit numbers:
There is only one, 10000 itself, and it must be included because there
are 10000 bicycles.

Total number of bicycle tags without 8's = 

8 + 8×9 + 8×9×9 + 8×9×9×9 + 1 = 8 + 72 + 648 + 5832 + 1 = 6561.

Total number of bicycle tags = 10000

Desired probability  = 0.6561



Edwin