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 (M-1) 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
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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)^(N-M)
Assume M≤N/2
So as M decreases by 1, the combination probability increases fivefold.

Part II:  Quantity of Combinations
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Number of combinations = {N! / [M! * (N-M)!]}

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! * (N-M)!]} = 2.5 * {N! / [(M-1)! * (N-M+1)!]}

Or

[(M-1)! * (N-M+1)!] / [M! * (N-M)!] = 2.5

Or

N-M+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 2017-05-25 11:42:49