Problem Statement
We throw 3 dice one by one. What is the probability that we obtain 3 points in strictly increasing order?
Solution
This is another puzzle that can be solved by applying the definition of probability (number of positive outcomes / total number of outcomes).
If we're throwing one die we have 6 possible outcomes, but we have 3 dice, so the total number of possible outcomes is 6^3.
We treat any 3 numbers in a strictly increasing order as a positive outcome of the experiment, so all three numbers have to be different.
For any three different numbers there's exactly one way to put them into increasing order, so we just need to calculate the number of combinations:
$$C_{6}^{3} = {6! \over 3!3!} = 20$$
Putting it all together we get our answer:
$$P = {C_{6}^{3} \over 6^3} = {20 \over 216} = 0.0926$$
Not convinced? Try the simulation below!
Simulation
show / hide simulation code
import numpy as np
from random import randint
nrounds = 10000
results = []
for j in range(nrounds):
trial_result = [randint(1, 6) for _ in range(3)]
results.append(trial_result[0] < trial_result[1] < trial_result[2])
simulated_probability = sum(results) / len(results)