The binomial distribution
When you repeat the same chance experiment a fixed number of times, how likely is each possible count of successes?
Where you are 4 inside
What this area is about
Many questions count successes: how many of 12 seeds sprout, how many of 40 guests fail to show. When there is a fixed number n of trials, each trial has two outcomes, the trials are independent, and every trial succeeds with the same probability p, the count X follows one pattern, the binomial distribution. One formula then gives every probability, with no list of the 2^n ways the trials can come out.
How these fit together
The setting. Recognizing a binomial setting checks those four conditions. Everything else here depends on them holding.
The formula. The probability of exactly k successes is P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}, and sums of it give “at most” and “at least”.
Center and spread. The binomial mean and standard deviation are \mu = np and \displaystyle \sigma = \sqrt{np(1 - p)}, with no table needed.
An approximation. The normal approximation lets a normal curve with that mean and standard deviation stand in for the bars when np \ge 10 and n(1 - p) \ge 10.
A path through it
In order: recognize the setting, then the formula for exactly k, then the mean and standard deviation, then the normal approximation, which uses both.
Finish here and the geometric distribution counts the trials until the first success, and the sampling distribution of a sample proportion builds on the same counts.