determining if it is convergent or divergent + taylor series
Hello Jt, since this is a wiki you don't need to address your questions to me. Anyone is free to answer them.
1. The integral from a to b over the function needs to equal 1. Hence A = 1/integral. See, e.g. here.
2. I'm not sure if I get where your confusion comes from. But a coin toss is the classical example of a Bernoulli distribution. Here p = 0.5, hence q = 1-p = 0.5. If you let Head be a success, then the probability of 3 successes is as asserted.
3. I'm not sure which question you are referring to. But if the question is something like it is twice as likely that roll a 6 than any other number you do the following: Let a be the probability of any other number. Then 2a is the probability of a 6. The total probability needs to be 1. With this information you can find a: a+a+a+a+a+2a = 1, i.e. a = 1/7 and thus prob(6) = 2/7.
4. The normalized binomial is needed to justify the use of a continuous random variable (with Gaussian as pdf) as an approximation to a discrete random variable (with binomial pdf). So in this sense it motivates chapter 8. In real life this is extremely useful in [Statistical hypothesis testing, but we didn't go into details here so I don't think you should worry about it.
5. Of course I can't comment on your first question. But yes, this is an example that, mathematically, it is the same as tossing a coin. But the application can we very different and easily applicable to a real world problem. Same with Hardy Weinberg genetics, since this relies on the binomial distribution (and is hence mathematically the same as a coin toss) you can apply the theoretical results on the binomial to genetics.
Hope this helps. Your questions are very welcome :)