Here's a problem I thought of that I don't know how to approach:

You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with significance level $\alpha $, where your null hypothesis is that the coin is fair and your alternative hypothesis is that the coin is not fair. In terms of $\alpha $, what is the expected number of flips before the first time that you reject the null hypothesis?

Edit based on comment below: For what values of α is the answer to the question above finite? For those values for which it is infinite, what is the probability that the null hypothesis will ever be rejected, in terms of $\alpha $?

Edit 2: My post was edited to say "You believe that you have a fair coin." The coin is in fact fair, and you know that. You do the hypothesis tests anyway. Otherwise the problem is unapproachable because you don't know the probability that any particular toss will come up a certain way.

You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with significance level $\alpha $, where your null hypothesis is that the coin is fair and your alternative hypothesis is that the coin is not fair. In terms of $\alpha $, what is the expected number of flips before the first time that you reject the null hypothesis?

Edit based on comment below: For what values of α is the answer to the question above finite? For those values for which it is infinite, what is the probability that the null hypothesis will ever be rejected, in terms of $\alpha $?

Edit 2: My post was edited to say "You believe that you have a fair coin." The coin is in fact fair, and you know that. You do the hypothesis tests anyway. Otherwise the problem is unapproachable because you don't know the probability that any particular toss will come up a certain way.