Stats problems - Probability
Hi, I've just started doing stats as at a-level I took all mechanics modules (stupid I know) and I'm having trouble getting my head around how to tackle questions of a probability nature. We use notation like P(AnB) etc, I'd be happy if anyone could show me how the solutions to these problems are worked out formally:
a) A student is to answer 7 out of 10 questions in an examination. How many choices are there? How many choices are there if at least 3 of the first 5 questions have to be answered? [Hint: split the exam into the first 5 and last 5 questions] I gather here we have to use the 'a choose b notation' but dont know where to start :S
b) A four character computer password is generated by choosing each character randomly from 26 letters (a,b,c,...,z) and ten digits (0,1,....,9). No password can contain only digits, and no password can contain the same character repeated four times. How many valid passwords are there?
c) A drug test for athletes is 95% effective in detecting a steroid when it is actually present. However, it also yields a false positive result for 1% of 'clean' athletes tested. If 0.5% of athletes use the steroid, what is the probability that an athlete who tests positive did actually use the drug?
I'm not asking for one person to do all the questions here, but it would be great if someone would be kind enough to do so. I'm trying to build up a collection of perfectly answered questions to help me gain a firm graps of the foundations of such questions, so any help would be vastly appreciated. thank you :)