MHB Probability of Rolling Sum of Die-Rolling

AI Thread Summary
The discussion explores the probability of obtaining specific sums from rolling a standard six-sided die multiple times. It establishes that the probability of rolling a sum of 2 is 7/36, while it is more likely to roll a sum of 6 than 1006. A formula is provided to calculate the probability of achieving a specific sum after n rolls, and it is noted that as n increases, the probability converges to approximately 2/7. For small values of n, the probabilities can be calculated easily, but they become more complex for larger sums. Overall, the analysis suggests that while certain sums are more likely, the probabilities stabilize around a mean value as the number of rolls increases.
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I ran into this problem, and would like to see if there is something more elegant.

Suppose we have a sequence $a_1, a_2, \dotsc, a_n, \dotsc$ where $a_k$
is the (running) sum of rolling a standard 6-side die $k$ times.

E.g. What's the chance of saying the number $2$ appears in this sequence? There are two possibilities, rolling a $2$ on the first roll, so $a_1 = 2$, or rolling two $1$'s, so $a_1 = 1, a_2 = 2$. Therefore, the probability is $\frac 16 + \frac 1{36} = \frac 7{36}$.

The question now is what is more likely to occur in this sequence? The number $6$ or the number $1006$?

There are formula to determine the probability of getting a sum of $t$ after $n$ rolls of a $6$-sided die:
\[
p = \frac 1{6^n} \sum_{k=0}^n (-1)^k \binom{n}{k} \binom{t - 6t - 1}{n-1}.
\]
We can further reduce the upper limit of the summation from $n$, and utilize code/Mathematica to evaluate this. If the code is carried out, I find that it is more likely you will see a $6$ than seeing a $2$ or $1006$.

Is there a "cleaner" way to show this without doing the actual arithemtics?
 
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magneto said:
I ran into this problem, and would like to see if there is something more elegant.

Suppose we have a sequence $a_1, a_2, \dotsc, a_n, \dotsc$ where $a_k$
is the (running) sum of rolling a standard 6-side die $k$ times.

E.g. What's the chance of saying the number $2$ appears in this sequence? There are two possibilities, rolling a $2$ on the first roll, so $a_1 = 2$, or rolling two $1$'s, so $a_1 = 1, a_2 = 2$. Therefore, the probability is $\frac 16 + \frac 1{36} = \frac 7{36}$.

The question now is what is more likely to occur in this sequence? The number $6$ or the number $1006$?

There are formula to determine the probability of getting a sum of $t$ after $n$ rolls of a $6$-sided die:
\[
p = \frac 1{6^n} \sum_{k=0}^n (-1)^k \binom{n}{k} \binom{t - 6t - 1}{n-1}.
\]
We can further reduce the upper limit of the summation from $n$, and utilize code/Mathematica to evaluate this. If the code is carried out, I find that it is more likely you will see a $6$ than seeing a $2$ or $1006$.

Is there a "cleaner" way to show this without doing the actual arithemtics?
Let $p(n)$ denote the probability that the number $n$ occurs in the sequence $\{a_k\}.$ I doubt whether there is a "clean" way to find $p(n).$ The mean score for a roll of the die is $\frac72$. So the average gap between consecutive terms in the sequence $\{a_k\}$ is $\frac72$. It follows that the asymptotic probability of a number occurring in the sequence is $\frac27 \approx 0.2857$. In other words, $p(n) \to \frac27$ as $n\to\infty.$ For small values of $n$, you can calculate $p(n)$ fairly easily. In fact, for $1\leqslant n\leqslant6$ the formula is $p(n) = \dfrac{7^{n-1}}{6^n}.$ When $n=7$ it is no longer possible to reach that score with one roll of the die, and the probability drops quite substantially (which I found quite surprising and counter-intuitive). After that, it gets rapidly more complicated to calculate $p(n).$ For $1\leqslant n\leqslant12$ the values of $p(n)$ (rounded to four decimal places) are $$\begin{array}{r|cccccccccccccc} n&1&2&3&4&5& 6&7&8&9&10 &11&12&\ldots & \infty \\ \hline p(n) & .1667 & .1944 & .2269 & .2647 & .3088 & .3602 & .2536 & .2681 & .2804 & .2893 & .2934 & .2908 && .2857 \end{array}$$ It looks as though $p(n)$ is going to converge quite rapidly to its asymptotic value $\frac27$, so I guess that $p(1006)$ will be very close to $.2857$, and will certainly be less than $p(6).$
 
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