Random Magic Paths - is it truly random?

[alexti]

Ivan Pedroso said:
"20 sages in a row - exactly 1 path missing":
(7/8)^20 *8 = 0.554

Explanation:
the (7/8) is the chance of getting one of the 7 "good" paths when you hire a sage.
^20 is just 20 such sages in a row.
*8 stems from the fact that you can choose your pool of 7 "good" paths from the 8 possible paths in 8 different ways.

7/8 probability means that you will get one of a "good" paths, but repeating the roll 20 times does not guarantee that you'll get each of "good" paths at least once. Thus 7/8 ^ 20 is probability of missing "bad" path (and maybe some of the "good" paths), not one path exactly. If you write your 20 random picks, using digits from 0 to 7 to indicate different paths, 8^20 is a total number of possible numbers (outcomes). Not getting one particular path means getting number without particular digits. There's total of 7^20 such digits. However, there're numbers like 111...1, 222...2 etc amongst them. Those numbers indicate outcomes where 7 paths are missing. Other numbers will represent outcomes with various number of paths missing.

Ivan Pedroso said:
General formula:
(chance of one good)^(number in a row) * (ways to make a good pool)

Getting 20 in a row without exactly 2 paths:
(6/8)^20 * 28 = 0.0888
(28=(8!)/(6!2!) is the number of ways of taking 6 from a sample of 8)

Same problem as above, it's probability of missing 2 paths or more.

Ivan Pedroso said:
Getting 20 in a row with 1 or more paths missing:
~65%

Surprisingly high ? or ?