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In the world of statistics, when we talk about the probability of success, we're dealing with how likely it is that a certain event will occur. To understand this, imagine flipping a coin. If we define 'success' as landing on heads, then the probability of success each time we flip the coin is 0.5, because there are two possible outcomes, heads or tails, each equally likely.

Now, consider the example of a binomial random variable with a probability of success represented as 'p'. In this case, we're looking at the chance that an event with two possible outcomes (success or failure) happens a certain number of times out of 'n' total attempts. For instance, if we're trying to find the chance of getting exactly 5 heads out of 10 coin flips, and each flip has a 0.5 probability of landing heads, this is where the binomial probability formula kicks in. Notably, the 'success' does not have to be something positive—it's simply the outcome we're measuring.

When solving problems involving binomial probability, we often come across the term binomial coefficient, represented by the symbol \( \binom{n}{k} \). It basically tells us the number of ways we can choose 'k' successes from 'n' trials. This is a fundamental aspect of combinatorics, a field of mathematics concerning the counting, combination, and permutation of sets of elements.

Mathematically, the binomial coefficient is calculated using factorials, which are denoted by the exclamation point (!) and mean to multiply a series of descending natural numbers. So, \( \binom{n}{k} \) is given by the formula \( \frac{n!}{k!(n-k)!} \). For example, if we are trying to find out how many different ways there are to get 3 heads (successes) from 8 coin flips (trials), we would compute \( \binom{8}{3} \) using the formula above. It’s like asking in how many different orders we could line up 3 heads and 5 tails, without worrying about which specific coin is which.

A binomial random variable represents the number of successes in a sequence of n independent experiments, or trials, each asking a yes/no question, and each with its own success probability, 'p'. What makes the variable binomial is the fixed number of trials and the fact that each trial has the same probability of success.

In our example, we have 8 trials (or flips of a coin), and each trial has a probability 'p' of resulting in success (getting a head). The binomial random variable 'X' would then be the total count of successes (heads) in these 8 trials. To calculate the probabilities associated with different potential outcomes of this random variable—like 'P(X ≤ 3)' which represents the probability of obtaining 3 or fewer successes—requires adding up the probabilities for all possibilities up to and including 3 successes. This is a classic illustration of how the binomial distribution works in probability and statistics.