Type-I vs. type-II error in statistical hypotheses testing Let us consider standard statistical hypotheses testing:
$$\alpha=P\{\text{type}-I  \text{ error}\}=P\{\text{Rejecting } H_0 \text{  when }H_0\text{ is true}\}$$
and
$$\beta=P\{\text{type}-II  \text{ error}\}=P\{\text{Accepting } H_0 \text{  when }H_1\text{ is true}\}.$$
My question is as follows: could you give an example of $\alpha$ and $\beta$
when tossing 10 coins so that I can see that it does not hold this equation:
$$\alpha=1-\beta.$$
More specifically, compose an outcome so that inequality between $\alpha$ and $1-\beta$ is seen.

Let’s say we’re testing a coin with probability $p$ of heads and our null hypothesis is $p=0$ (that it never comes up heads. We flip 10 times and our procedure is that we reject if we see a head, otherwise we accept. 
If the null hypothesis is true we will definitely not see a head, so $\alpha=0.$ However if not and we have $p>0,$ then the probability of 10 tails in a row is $(1-p)^n,$ so we have $\beta =(1-p)^n\ne 1-\alpha.$
There is no reason whatsoever to expect $\alpha=1-\beta.$ Usually, since the null hypothesis is a simple hypothesis, $\alpha$ is just a number, whereas since the alternative hypothesis is compound, $\beta$ depends on the effect size. So there’s usually a particular effect size for which they’re equal (in this silly example it’s $p=0$), but there is no real significance to this value. (One would hope they were both small for effect sizes of interest. Traditionally, the rule of thumb is to fix 5% for $\alpha$ and then set sample size so that $\beta$ is less than 20%.)
