The true proportion of respondents

Fellow, stats lovers!

I am now trying to remember my early statistics course in order to solve the following problem:

1) During the survey, 40% of the population maintain interest in mathematics. 1000 people were interviewed. With what probability can it be argued that the proportion of respondents who support mathematics differs from the true proportion by no more than 0.05?
Should I use this statistical model: P((μn/n)−P≤0.4)≥Z ?

2) Is it more reasonable to use the Bayes' theorem or the full probability formula to solve the following problem: A product defect is 5%. Each product with the same probability can be checked by one of 2 controllers. The first of them detects an error with a probability of 0.7, the second - with a probability of 0.8. What is the probability that a recognized product is defective?

Thank you in advance for your assistance!
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2) is going to bayes. Probability of recognizing given defective will be .5(.7 + .8). Question is asking for Probability of defective given recognized as such. Seems like you might be missing a piece of information though...