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  • Suppose I am playing a game where I flip a $k$ sided coin (i.e. sides are denoted by $k_1$, $k_2$ ...$k_k$) .
  • There is a probability of $p_1$, $p_2$, ...$p_k$ of the coin landing on any one of these sides
  • A score of $c_1$, $c_2$, ...$c_k$ associated with coin landing on each side ($c_i$ can be positive or negative integers)
  • At each turn, my score becomes: current_score + $c_i$
  • I play this game for $n$ turns
  • The result of each turn is independent from the previous turn

My Question: If my current score is 0 , after 5 more turns - I want to know how many outcomes can happen (e.g. $x$ number of games games where my score is 5*$c_1$, $y$ number of games where my score is 4*$c_1$ + $c_2$, etc.), and the probability of obtaining each one of these combinations (e.g. there is a probability of $q_1$ where I end up with a score of 5*$c_1$, etc.)

My attempt to solve this problem: I think that this question can be answered with the Multinomial Distribution. Here, $x_i$ is the number of times that the coin landed on face $k_i$ and $p_i$ is the probability of the coin landing once on face $k_i$. Thus, the probability of getting any $n$-length sequence in any order is given by: (this is the equivalent of saying that, what is the probability in $n$ flips that the coin lands $x_1$ times on face $k_1$, $x_2$ times on face $k_2$ ... and $x_k$ times on face $k_k$)

\begin{align} f(x_1,\ldots,x_k;n,p_1,\ldots,p_k) & {} = \Pr(X_1 = x_1 \text{ and } \dots \text{ and } X_k = x_k) \\ & {} = \begin{cases} { \displaystyle {n! \over x_1!\cdots x_k!}p_1^{x_1}\times\cdots\times p_k^{x_k}}, \quad & \text{when } \sum_{i=1}^k x_i=n \\ \\ 0 & \text{otherwise,} \end{cases} \end{align}

For example, here is the probability of the coin landing 5 consecutive times on $k_1$ (in any order) :

$$P(X = k_1, k_1, k_1, k_1, k_1) = \frac{5!}{1!1!1!1!1!} p_1^{1} p_1^{1} p_1^{1} p_1^{1} p_1^{1}$$

And here is the probability of the coin landing 3 times on $k_1$ and 2 times on $k_2$ in $n = 5$ turns (in any order):

$$P(X = k_1,k_1,k_2, k_2, k_2) = \frac{5!}{2!3!} p_1^{2} p_2^{3}$$

In general, after 5 turns, there can be $5Ck$ score combinations: $$5Ck = \frac{5!}{k!(5-k)!}$$

From here, I would have to identify which combinations I am interested in. For example, suppose I am interested in a score of $b$ after 5 turns. I would identify which of the $5Ck$ combinations sum to $b$ (e.g. perhaps $c1 + c3 + c9 - c2 + c5 = b$ , perhaps $c10 + c3 + c9 - c2 + c5 = b$, etc). I think these combinations can be counted like this (see references):

$$\text{No. of ways}= {b - 1 \choose 5-1} = \frac{[b-1]!}{[5-1]![b - 1 - (5 - 1)]!} $$

I would then need to find a formula to identify each combination that result in a sum of $b$ after $n$ = 5 turns and weigh each combination by its corresponding probability .... and then sum each weighted combination.

  • Is there such a compact formula I can use to solve my original question?
  • In general, is my analysis correct?

Thanks!

References:

stats_noob
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    It's more complicated than that, since different combinations may lead to the same tally, depending on the $c_i$ values. For instance, say that $k=3$ and $c = 1, 2, 3$, you may obtain a score of 5 with $1, 1, 3$ but also with $1, 2, 2$. I don't think there is a shorter way than enumerating all the possibilities and then aggregate the outcomes with the same tally. – nicola Nov 17 '23 at 05:59
  • @nicola: thank you for your reply! I will keep looking into this. lately I have become very interested in these types of problems. I can't stop thinking about them! – stats_noob Nov 17 '23 at 07:50
  • Here are some other problems I thought about: https://math.stackexchange.com/questions/4804927/calculating-the-probabilities-of-a-100-sided-weighted-dice , https://math.stackexchange.com/questions/4804059/how-to-tell-if-a-problem-has-no-analytical-solution – stats_noob Nov 17 '23 at 07:51
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    The pattern of values C_i is very important in your problem. if $C_0=1, C_1=2, C_k= 2^k$, the problem is simple, you have no duplicates, for any tally, there is only 1 way to pbtain this value. If $C_k=k$ (like a standard 6-faces die), it is much more complex, because you have many ways to obtain medium values. – Lourrran Nov 17 '23 at 15:45

1 Answers1

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Maybe we should begin with the basic:

$$ (1+x+x^2+x^3.... +x^k)^n $$

Is there a formula to calculate the coefficient of $x^a$ (where $a$ can be any integer value less than k^n) that's more efficient than grinding through every possible permutation that sums to $a$ with the multinomial theorem and then summing them all?

Older Amateur
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  • Should be a comment, but i don't yet have 50 rep. – Older Amateur Nov 17 '23 at 17:21
  • Seems like this question has been asked here. https://math.stackexchange.com/questions/3656079/coefficient-of-xi-in-xx2-xkn?noredirect=1 Euler has a paper on it :) https://arxiv.org/pdf/math/0505425.pdf – Older Amateur Nov 17 '23 at 19:17
  • thank you so much for your answer! I can not believe that the question I thought of was also thought of by Euler! – stats_noob Nov 23 '23 at 18:58