The following problem is something I have come up with recently, and have been trying to solve, but to no avail. I wonder whether the solution is relatively easy, or pretty complicated:
Suppose we have an unfair die, such that the probability to get the number $s\in\{1,2,3,4,5,6\}$ is $p$, where $p>1/6$ and $s$ is deterministic (but unknown). We call $s$ the special number. Suppose that the probability to get an element of $\{1,2,3,4,5,6\}\setminus\{s\}$ (i.e., not a special number) is $(1-p)/5$. We denote $X_n$ to be the result of the $n$-th toss. Let $N\in\mathbb{N}$, and suppose we know the results of first $N$ tosses: $\{X_n\}_{n=1}^{N}$.
Given the first $N$ results, we want to guess (estimate) the special number $s$. We do this with a majority vote: We pick the mode of the sequence $X_n$. In other words:
$$\hat{s}=\underset{k\in\{1,2,3,4,5,6\}}{\mathrm{argmax}}\left\{\sum_{i=1}^{N}1_{\{X_i=k\}}\right\}$$
Where $1_{\{A\}}$ is the indicator function of the event $A$. What is the probability that $\hat{s}\neq s$ in terms of $N$ and $p$?
- You can assume that there isn't a tie in the majority vote (meaning, $\hat{s}$ is defined well).
The only thing that I am certain of is:
$$\mathbb{P}\{\hat{s}\neq s\}=1-\mathbb{P}\{\hat{s}=s\}=1-\frac 16\sum_{k=1}^{6}\mathbb{P}\{\hat{s}=k|s=k\}=1-\mathbb{P}\{\hat{s}=6|s=6\}$$
This is due to symmetry.