How to calculate the probability of rolling matching dice?

Since you mention doing this with Python simulations, I will go down that route. Here is the code that we will use which only uses the built in Python libraries random and collections. The key function is `generate_frequencies`, which takes the number of dice you want to roll and then the total number of replications for the simulation (default to 10000).



from collections import Counter
import random

def generate_rolls(num_dice, num_rolls, num_sides = 10):
    outcomes = []
    for _ in range(num_rolls):
    outcomes.append([random.randint(1, num_sides) for _ in range(num_dice)])
    return outcomes

def num_matches(outcome):
    counter = Counter(outcome)
    return sum(x for x in counter.values() if x > 1)

def generate_frequencies(num_dice, num_rolls = 10000, num_sides = 10):
    outcomes = generate_rolls(num_dice, num_rolls, num_sides)
    match_counts = Counter(num_matches(out) for out in outcomes)
    return {outcome: match_counts[outcome]/num_rolls for outcome in range(num_dice+1)}




As an example, here is how you can calculate the probabilities of rolling $n$ matches from rolling 6 ten-sided dice:

print(generate_frequencies(6))
{0: 0.1513, 1: 0.0, 2: 0.4466, 3: 0.1011, 4: 0.2433, 5: 0.0448, 6: 0.0129}
 

By increasing the num_rolls parameter from its default of 10000, you can get more accurate estimates of the probably, although the default should be fairly adequate. Also, you can change the num_sides parameter for different sided dice, e.g. generate_frequencies(2, num_sides=8) will calculate the probably of 0 or 2 matches for 2 eight-sided dice.

From here you can take this code and plug and play with different numbers to generate the probabilities you are interested in.