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Probability is a fundamental concept in statistics that measures the likelihood of an event happening. When dealing with a binomial distribution, probability helps quantify scenarios where there are two possible outcomes: success and failure. For example, in our motel problem, when someone inquires about a room, either they decide to rent a room or not.
The probability of a single person renting a room is 40% or 0.4. This probability can tell us how often we can expect a particular outcome in repeated trials under identical conditions.

• Success probability (\(p\)) is 0.4, meaning the person rents a room.
• Failure probability (\(1 - p\)) is 0.6, meaning the person does not rent a room.

By understanding the probability of both renting and not renting, one can calculate how many inquiries are needed to reach a certain level of certainty, like the 99% mentioned in the problem.

The expected value is a critical concept in probability that provides the average outcome if a certain action is repeated multiple times. It's like the long-term average, indicating what you can "expect" to happen. In the context of our motel problem, the expected value helps us determine how many room rentals will likely result from inquiries.
The formula for expected value in a binomial distribution is given by \(E(X) = np\), where \(n\) is the number of trials (or inquiries), and \(p\) is the probability of success (a room rental).

• If \(n = 25\) and \(p = 0.4\), then \(E(X) = 25 \times 0.4 = 10\).

This means that out of 25 inquiries, we can expect, on average, about 10 people to rent a room. The expected value doesn't guarantee precise numbers every time, but rather an average over lots of trials.

Random variables are used to map outcomes of random processes to numerical values. They help us quantify randomness and are essential for calculating probabilities and expected values. In the binomial distribution, a random variable is defined to represent the number of successes in a fixed number of trials.
In our motel scenario, let \(X\) be the random variable representing the number of room rentals. Here \(X\) can take any value from 0 to the total number of inquiries, recording how many of these inquiries result in a rental.

• A discrete random variable can only take specific values (like our \(X\)), counting the number of successes.
• Each outcome (rental or no rental) contributes to the overall distribution of \(X\), informing probabilities and expected values.

Understanding random variables allows us to model and work with real-world scenarios statistically.

The complement rule is a handy tool in probability that helps calculate the chance of at least one success. Since directly calculating \(P(X \geq 1)\) can sometimes be complex, the complement rule simplifies it by calculating the likelihood of the opposite event and then subtracting from 1.
For example, \(P(X \geq 1) = 1 - P(X = 0)\). This tells us the chance of getting at least one success is the total probability (1) minus the probability of zero successes.

• For the motel case: \(P(X = 0) = (1-p)^n\), where all inquiries fail to result in a room rental.
• If \(P(X = 0) = 0.01\), then \(P(X \geq 1) = 1 - 0.01 = 0.99\), confirming the 99% surety of at least one rental.

Using the complement rule greatly simplifies solving complex probability questions and is especially useful in binomial distributions.