Problem:
You place three dots along the edges of an octagon at random. What is the probability that all three dots lie on distinct edges of the octagon?
This is a very easy problem, which we can solve computationally:
import itertools
generator = itertools.product([1,2,3,4,5,6,7,8], repeat=3)
valid = 0
total = 0
for i in generator:
if i[0] != i[1] and i[1] != i[2] and i[0] != i[2]:
valid += 1
total += 1
print(valid / total)
0.65625
This gives us a probability of 0.65625.
If we want to do this mathematically, we first need to count the total number of ways to place 3 dots along the edges of the octagon which is $8^3 = 512$ since for every dot we have 8 choices each time.
Then we need to count the number of ways to place 3 dots along the edges of the octagon where all 3 dots are on distinct edges which is $8 \cdot 7 \cdot 6 = 336$ since for the first dot we have 8 choices, for the second dot we have 7 choices, and for the third dot we have 6 choices.
Therefore, the probability is $\frac{336}{512} = 0.65625$.