(1) looks good.

With (2) you start off "since is divisible by 4" but that is what you are trying to prove. You have to start by proving it is true for n= 1 (which is easy) andthensay "If, for some k, is divisible by 4" or "supposeis divisible by 4". The difference is that (1) I am saying "if" rather than "since" and (2) I am using "k" rather than "n". That is not crucially important but makes it clear that I amnotassuming what I want to prove.

For (3), I personally don't like using "formulas" like that. Here is what I would do:

Assume a population of 10000 people. 4%, 400, use drugs and 96%, 9600, do not. There is a test for drug use which has a 3% false positive rate and a 2% false negative rate. So of the 9600 who do not use drugs, 97%, 9312, will test negative and 3%, 288, will test positive. Of the 400 who use drugs, 98%, 392, test positive and 2%, 8, test negative.

So there are a total of 288+ 392= 680 who test positive, 392 of whom are really drug users. So the probability that a person who tests positive really is a drug user is [tex]\frac{392}{680}= .5765... or 57.65% as you have.

Similarly, there are a total of 9312+ 8= 9320 who test negative, 9312 of whom do not use drugs. So the probability that a person who tests negative is not a drug user is or 99.9% as you have.