False positive rate and false negative rate on testing the fairness of a number of coins

Question

I faced a problem which can be described as follow:

There are 9 independent coins that we don't know if each is biased or no. One can toss them individually and use binomial test to test if it is biased.

Let's assume $\alpha$ for false positive rate and $\beta$ for false negative rate. In the first tossing round, one finds that there are 4 hypothesis tests are significant, and conclude 4 "biased" coins after tossing them for enough times respectively. Then one selects these 4 "biased" coins, and run the 2nd tossing round, now one gets 3 hypothesis tests are significant. Finally, one concludes that these 3 coins are biased.

So the question is, given the above scenario, what is the overall false positive rate and false negative rate?

My approach for calculating false positive rate is

$\binom 94$$\alpha$⁴ $\binom 43$$\alpha$³

Similarly, for false negative rate is

$\binom 94$$\alpha$^{9 - 4} $\binom 43$$\beta$^{4 - 3}

I wonder if my approach is correct or not. Thanks for any comments!