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Understanding probability theory is essential for tackling problems involving likelihoods, like those in binomial distributions. Probability provides us with a way of quantifying the chance of an event happening. If you throw a fair six-sided die, the probability of any given number turning up is \( \frac{1}{6} \). The same logic applies when calculating the probability of a certain number of airline passengers choosing to sleep.
Probability theory involves:

• Defining possible outcomes.
• Assigning probabilities to these outcomes.
• Calculating the likelihood of events using various probability rules and theorems.

In the context of our problem, the probability formula for a binomial distribution helps in determining how likely it is for a certain number of people to choose to sleep. This is governed by known parameters like the total number of passengers and the probability that a single passenger will sleep. This approach makes use of probabilities calculated from the uniform responses of surveyed passengers, translated into predictions for different scenarios.

In probability, a random variable is a way to map outcomes from a chance experiment to numerical values. In simpler terms, it's a function that gives us a number to represent the outcome of a random process.
For example, when tossing a coin, you might let "heads" represent 0 and "tails" represent 1. Random variables can take on several values, and the likelihood of each value arising can typically be represented by a probability distribution.
In our specific airline example, the random variable is "\(x\)", denoting the number of passengers out of six who choose to rest or sleep. Because the number of people choosing to sleep is random—though it has known probabilities—the outcomes are modeled using the concept of a random variable.
Here, \(x\) follows a binomial distribution, where the number of trials is 6 and the probability of "success", defined as a passenger sleeping, is 0.8. The binomial distribution is particularly useful when each trial is independent, and there is a fixed probability for each trial, both of which apply to our flight scenario.

Combinatorics is a branch of mathematics focused on counting and arranging. It plays a vital role in probability theory, especially in calculating probabilities in binomial distributions where combinations come into play.
To find out probabilities like \(P(4)\), the binomial coefficient \( C(6,4) \) is used. This coefficient tells us how many ways we can choose 4 passengers out of 6 to rest or sleep, without considering the order. It can be calculated using the formula:
\[ C(n, k) = \frac{n!}{k!(n-k)!} \]
where \( n! \) denotes factorial of \( n \). In the airplane exercise, this calculation allows us to calculate how many ways passengers can arrange themselves into sleepers and non-sleepers.
Thus, combinatorics provides a systematic way of counting possible configurations and is deeply linked to probability distributions.