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Probability theory is the mathematical framework that helps us understand the likelihood of different outcomes in situations with uncertainty. In the context of a Myrtle Beach resort hotel, it allows us to explore how likely it is for certain numbers of rooms to be occupied.

When dealing with probabilities, we consider the possible outcomes and assign them a value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

• If something is likely to happen, its probability is closer to 1.
• If it's unlikely, the probability is closer to 0.

For example, finding the probability that at least half of the hotel rooms are occupied involves calculating the likelihood of having 60 or more rooms filled out of 120.

The key is to use tools like the cumulative distribution function, which helps us sum probabilities over a range of outcomes. Understanding these concepts is foundational to applying probability theory effectively in statistical calculations.

Random variables are a core component of probability theory. A random variable is a variable that can take on different values, each with an associated probability. In our hotel room problem, the random variable, denoted as \(X\), represents the number of occupied rooms on a given day.

It is important to understand the type of distribution that a random variable follows.

• Here, \(X\) follows a binomial distribution, characterized by a fixed number of trials \(n\), which is 120 in this case, and the probability of success, which is the probability that any given room is occupied, \(p = 0.75\).
• The binomial distribution is used when there are two possible outcomes for each trial (e.g., a room is either occupied or not).

Recognizing the distribution of a random variable allows us to make precise probability calculations, such as determining the probability that at least 100 rooms are occupied or that 80 or fewer rooms are filled.

Statistical calculations involve methods and procedures used to analyze data and quantify uncertainty in various scenarios. In our exercise with the hotel, we perform several statistical calculations to understand room occupancy probabilities.

To solve these problems, we use tools such as:

• **Cumulative Distribution Functions (CDFs):** These functions help determine the probability that a random variable is less than or equal to a certain value. For example, to calculate the probability that 80 or fewer rooms are occupied, we use a CDF.
• **Complement Rule:** Often, we calculate the complement of a probability to simplify calculations. For instance, to find the probability of at least 60 rooms being occupied, we calculated \(1 - P(X < 60)\).

Using these streamlined statistical methods makes complex calculations manageable, especially when dealing with binomial distributions in large datasets like our 120-room hotel scenario.