Dice Problem

Suppose you have a singular die and you roll it. Then you roll the number of dice that show on the last roll you made. Repeat this until you have rolled at least 100.

What numbers do you expect not to roll during the last round?

In [58]:
import random as r
from pylab import *
from numba import njit
In [59]:
@njit
def rolls():
    numDice = 1
    while numDice < 100:
        newNumDice = 0
        for i in range(numDice):
            newNumDice += r.randint(1,6)
        numDice = newNumDice
    return numDice
In [65]:
def run(trials):
    numDiceDict = {}
    for _ in range(trials):
        numDice = rolls()
        if numDice in numDiceDict:
            numDiceDict[numDice] += 1/trials
        else:
            numDiceDict[numDice] = 1/trials
    print("number rolled on the last round vs. frequency")
    bar(list(numDiceDict.keys()), list(numDiceDict.values()));
In [83]:
run(10000)

number rolled on the last round vs. frequency
In [85]:
run(10000)

number rolled on the last round vs. frequency
In [86]:
run(10000)

number rolled on the last round vs. frequency
In [87]:
run(10000)

number rolled on the last round vs. frequency
In [89]:
run(10000)

number rolled on the last round vs. frequency
In [93]:
run(10000)

number rolled on the last round vs. frequency
In [95]:
run(10000)

number rolled on the last round vs. frequency

Why are the bands different?!

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