Binomial probability theory and how do we determine "p" value?

Answer 1

In binomial probability we repeat a single experiment in n independent trials. We categorise the results as "success" or "failure", where the probability of success in each trial is a constant, p. Usually p is known and we have to determine the probability of having a certain number r of successes.

The probability of getting r successes in n trials is given by the formula C(n, r).p^r.(1-p)^(n-r). Of course success and failure are arbitrary; you can reverse the categories and replace p with 1-p and r with n - r and you will have the same results.

Having r successes in n trials is the same as having n-r failures, each with probability (1-p); in this case the formula becomes C(n, n-r).(1-p)^(n-r).(1-(1-p))^(n-(n-r)) = C(n, n-r).(1-p)^(n-r).p^r which is the same as the original since C(n, r) = C(n, n-r).

For example, if you throw a standard die 100 times, the probability of getting (exactly) 10 sixes is C(100, 10).(1/6)^10.(5/6)^90 = 2.14% to 3 d.p.

Answer 2

If you are sampling from a large population, you can never know the p-value, you can only estimate it to a certain confidence. Your estimate of p will be between confidence limits that are a function of your sample size and the confidence you desire. The formula for these limits is:
r/n +- z(sub alpha/2)(sqrt((r/n)(1-r/n)/n)
You can state that the estimate of the population p will lie between these limits with 1-alpha confidence.
Alpha is the probability (risk) that the actual (but unknown) p-value does not lie within your calculated limits. You want this to be small.
z(sub alpha/2) is found in the standard normal table. For example, if you want to estimate p with 95% confidence, then alpha = 0.05 and z(sub alpha/2) is +-1.96.
r is the number of successes.