In
"5 dice" Andy had five regular dice. Now he has a total of N regular dice. He claims that the odds of rolling exactly M sixes is exactly half as likely as rolling (M1) sixes. (M < N).
For what values of N is this true?
State the pattern if there is one.
Express M as a function of N.
This equation is divided into two parts.
Part I: Probability of Combination

Chance of rolling a 6 on one die = (1/6)
Chance of not rolling a 6 = (5/6)
Probability of a combination = (1/6)^M * (5/6)^(NM)
Assume M≤N/2
So as M decreases by 1, the combination probability increases fivefold.
Part II: Quantity of Combinations

Number of combinations = {N! / [M! * (NM)!]}
So the key here is to find combination values where the value above is 2.5 times greater than its value when M decreases by 1.
{N! / [M! * (NM)!]} = 2.5 * {N! / [(M1)! * (NM+1)!]}
Or
[(M1)! * (NM+1)!] / [M! * (NM)!] = 2.5
Or
NM+1 = 2.5*M
Or
M = (N+1)/3.5
Since we are dealing with whole numbers, N+1 must be divisible by 7.
Starting with N=6, M=2
N=13, M=4
N=20, M=6
N=27, M=8
And so on.

Posted by hoodat
on 20170525 11:42:49 