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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.

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

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    Hello! Can you be more accurate about the goal of your test and the assumptions you are doing? For example, are you getting $n$ heads or is the $n$th flip a head? If you took this exercise from a textbook or just any available source, it could help you (and whoever will answer your question) if you cite the whole text here. :-) – 2017-01-18
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    Hi and thanks for comment. This is not a homework or is not any exercise from any textbook. It is some part of my research and I changed the problem to simpler one that is flipping a coin. we see n times H after n times flipping. The coin is biased but every time it biasedness may be different. But based on the results so far we want to guess that if the n+1 th time should we expect H again or not. – 2017-01-18

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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.

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    Thank you for your answer. But that α is the H0 rejection probability not the probability of showing up head. We do not know exactly what is the probability of showing up head. But we want to know based on the n times Head should we expect another head at n+1 time of flipping or not? – 2017-01-18
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    @SSD So you want a yes or no guess for whether or not the (n+1)-th coin flip will be a head? I would think that an estimated probability would be more useful, but if you want a yes/no guess then you can just say that you would expect another heads is the estimated probability of getting heads is at least 50%. Hypothesis testing still doesn't make sense because the $n+1$th coin flip is a random variable, not a parameter. – 2017-01-18
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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: $$ \begin{equation} \begin{cases} H_0 \colon p \le p' & \\ H_1 \colon p > p' & p' \in (0,1) \end{cases} \end{equation} $$

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    The p' value is not important in this problem. What that is important is that how much I can be sure about the next flip Showing up Head. As I said the p' is not just one value and can be different each time we flip the coin – 2017-01-18