Discrete random variables occur in situations where we can list every possible outcome of a process, and assign each outcome a probability in a probability model. For example, how many heads will we get in three flips?
Expected value gives us the mean or long-term average number of success we can expect over time if we repeat a process over and over. For example, how much money can someone expect to win on average for each lottery ticket they buy?
Binomial random variables are a special type of variable that comes up we repeat a process for a set number of independent trials, with each trial having the same probability of success, and we count how many successes we get at the end of the trials. For example, how many shots will a basketball player make in a series of 10 shots?