Sum = N

Probability Level pending

You roll a fair six sided die continuously until the total equals $N$ (or greater) and then stop.

What is the probability that when you are done, that the total will equal exactly $N$ for very large $N$ .

i.e.

Let $P(N) =$ Probability that eventually the total will equal exactly $N$ .

If:

$\lim_{N \rightarrow \infty} P(N) = \frac{a}{b}$

where $a$ and $b$ are coprime numbers. What is $a+b$ ?

1 solution

Geoff Pilling
Oct 23, 2016

For $N=1$ , clearly $P(1) = \frac{1}{6}$ .

For $N=2$ , you can roll a $2$ with a $\frac{1}{6}$ chance, or two $1$ 's with a $\frac{1}{36}$ chance.

$P(2) = (1/6)*P(1) + 1/6 = 7/36$

In general,

$P(N) = (\frac{1}{6})(P(N-6)+P(N-5)+P(N-4)+P(N-3)+P(N-2)+P(N-1))$

Where $P(0) = 1$ , and $P(N) = 0$ for $N<0$

Solving,

$\lim_{N \rightarrow \infty} P(N) = \frac{2}{7}$

$2+7 = \boxed9$

I think you should provide more explanation for the equations. Specifically, explain that you are conditioning on the very last total.

Also, can you explain why the limit is $\frac{2}{7}$ ? It seems to work out numerically, but is there a reason for the exact value?

Calvin Lin Staff - 4 years, 7 months ago