Question

For two independent hypothesis testing, if the significance level are both set to be 10%.
What is the probability that we make at least one mistake, if the null hypotheses are both true?

Answer 1

You did not phrase the question as clearly as you ought to have, however I will take a swing at what I am interpreting your question to be.
A significance level for the hypothesis H being true of 10% typically means that the hypothesis is true with 90% confidence. If we have evaluated two separate hypotheses and determined each to be true at the 10% confidence level, then the probability that at least one is actually false is equal to the probability that the first one is false and the second true, plus the probability that the second is true and the first false, plus the probability that both are false. If we work this out, we get: 0.9*0.1+0.9*0.1+0.1*0.1=0.19. Notice that this is equal to 1 minus the probability that but hypotheses were true, which looks like 1-0.9*0.9=0.19.