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Alright, so I'm trying to figure out the expected value of dice throws in a game I'm making, because I would like for the options the player has to be statistically balanced.

A person, for their special attack, can roll four dice. However, if their opponent's special defense is in the same slot, they can remove the highest three of those dice, and I would like to know the expected value of this action: removing the three highest dice from four. (I need this for the rest of my balancing).

I saw this: The expected payoff of a dice game

And this: Expected Payoff for Dice Game Where Six = No Payoff

But don't think they necessarily apply here, or I can't figure out how to make them apply to this problem.

I knocked up a simply python simulation that does this for me, and got a mean of 1.755 over a hundred million trials, but I like statistics, and would like to know why this is the result.

import random
import statistics

def Rand(start, end, num):
    res = []   
    for j in range(num): 
        res.append(random.randint(start, end)) 
    return res

all_list = []

for i in range(100000000):
    take = Rand(1,6,4)
    take.sort()
    all_list.append(take[0])

print(statistics.mean(all_list))
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  • $\begingroup$ $(1\times6+(2^4-1^4)\times5+(3^4-2^4)\times4+(4^4-3^4)\times3+(5^4-4^4)\times2+(6^4-5^4)\times1)/6^4 \approx 1.755$. $\endgroup$ – Hw Chu May 7 at 16:04
  • $\begingroup$ @HwChu Please post that as an answer, with at least a brief explanation of the reasoning. $\endgroup$ – Ethan Bolker May 7 at 16:07
  • $\begingroup$ To see why, for instance, the possibility that you get a 3 is when the least of the four dices is exactly 3. The number of combinations that all four dices are at least 3 is $4^4$, while the number of combinations that all four dices are at least 4 (which you shall exclude) is $3^4$, so the probability that the least dice is 3 is $(4^4-3^4)/6^4$. $\endgroup$ – Hw Chu May 7 at 16:12
  • $\begingroup$ Ah okay, thank you. I thought it would probably have something to do with enumerating all the possible states, but didn't think to do it this way. $\endgroup$ – CapnShanty May 7 at 16:41
  • $\begingroup$ See en.m.wikipedia.org/wiki/…. $\endgroup$ – amd May 7 at 23:58

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