Estimation of $n+1$-th coin toss, assuming $n$ heads on previous tosses, and coin is unbiased.

Question

Suppose we have a coin and it is biased but the amount of its biasedness is itself a probability and we do not know it exactly. If we flip coin $n$ times and we get $n$ heads, what is the probability that we see the HEAD in $(n+1)$th time of flipping the coin? In other words, can we make sure that based on some probability $\alpha$ of rejecting $H_0$, it will be HEAD again?

I am trying to solve this with Hypothesis testing.

$$\begin{equation*} \begin{cases} H_0 \colon X_{n+1} = H \\ H_a \colon X_{n+1}= T \end{cases} \end{equation*}$$

Answer 1

If I understand correctly, you have a coin with some probability $\alpha$ of showing up heads and some probability $1-\alpha$ of showing up tails, and you want to estimate $\alpha$ based on the first $n$ flips? This would be an estimation problem, not a hypothesis testing problem. There are many ways to develop estimates. The simplest would be just to use the proportion of flips that showed up heads in the first $n$ flips. That would be a consistent, unbiased estimator.

Answer 2

Maybe hypothesis testing isn't the appropriate way to get to the result. Usually, hypothesis tests pose questions about the distribution of a population - in this case, it is totally sum up by $p \in (0,1)$ - rather than dealing with possible outcomes. What you can do is to set a test about the bias of your coin: $$ \begin{equation} \begin{cases} H_0 \colon p \le p' & \\ H_1 \colon p > p' & p' \in (0,1) \end{cases} \end{equation} $$