$How to calculate the probability of rolling matching dice?$

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$How to calculate the probability of rolling matching dice?$

Hello!

I'm trying to design a new dice system for a table-top roleplaying game. I think that a multinomial distribution or some sort of simulation (probably 100,000 simulations of rolling 10-sided dice) might be the right way to go, but I need some help with the statistics and coding up a solution in python.

In the game, the player will roll a number of ten-sided dice. Every dice that matches the value of at least one other dice will count as a "match". If a player rolls 8 dice with values {1, 2, 3, 3, 3, 4, 4, 5}, they will have 5 total "matches" because there are three 3s and two 4s. I've been able to intuit that if the player rolls only 2 dice, they would expect to have a 90% chance of no matches, and a 10% chance of 2 matches (one pair of matching dice).

At some point I expect more dice to mean a lower chance of getting "n" matches -- for example, if you rolled 12 dice it would be impossible to get less than 3 matches {1,2,3,4,5,6,7,8,9,10,1,1}.

I'd like to be able to generate probabilities of getting "n" matches with a specified number of same-sided dice (i.e. rolling a number of 8-sided dice, a number of 10-sided dice, or a number of 12-sided dice). I'd like to be able to specify the # of matches I want a probability for, the # of dice to be rolled, and the # of sides on the dice rolled (all dice rolled can have the same # of sides).

Thank you!

Probability Probability Distribution

Fxfaust

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Mathe

We could solve this but I suggest increasing the bounty to $25

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Mathe

We could solve this but I suggest increasing the bounty to $25

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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.

Amas

Fxfaust

Thank you! I'll give this a try soon.

$How to calculate the probability of rolling matching dice?$

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