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Probability measures the likelihood of an event occurring. It's a fundamental concept in statistics used to predict outcomes. For instance, if you have a 40% chance of renting a room to an inquiring customer, in probability terms, this would be written as a probability of 0.4. In exercises, you'll often see scenarios asking "what are the chances" of an event, such as renting a room.

• Probability of an event = Number of favorable outcomes / Total number of possible outcomes
• The probability is always a number between 0 and 1
• A probability of 0 means an event will not happen, and a probability of 1 means an event will certainly happen

Using probability helps us make informed predictions based on quantitative data, especially when dealing with uncertainty, like deciding how many inquiries are necessary to ensure a high likelihood of room rentals. Breaking down real-world problems into simple probability terms makes them easier to solve.

Expected value gives us the average outcome from a set of possible outcomes, each with a specific probability. It's like the long-term average you'd expect from a random process if you repeated it many times. To calculate the expected value, multiply each possible outcome by its probability and then sum these results. In our motel scenario, each room inquiry has a 40% chance of resulting in a booking. If 25 people ask about rooms, we compute the expected number of bookings as follows:

• Formula: \(E = n \times p\)
• For 25 inquiries and a 40% rental rate: \(E = 25 \times 0.4\)
• This results in an expected value of 10 bookings

The expected value helps businesses anticipate outcomes and manage resources effectively, since they can predict how many rooms they might rent on average given historical inquiry data.

The complement rule is a handy tool in probability, helping us find the probability of an event happening by looking at the probability of its complement (the event not happening). It's based on the principle that either an event occurs or it doesn't, encompassing all possible outcomes. The complement rule is crucial when calculating probabilities like needing a 99% chance to rent at least one room.

• The probability of an event = 1 - Probability of its complement
• In probability terms, complement of an event \(A\) is represented as \(A^c\)
• For example, if the chance of not renting a room from one inquiry is 0.6, then the chance of renting is 1 - 0.6

In practice, the complement rule simplifies problems like how many inquiries you need to ensure high booking probabilities by considering what would happen if none resulted in bookings.

Logarithms are mathematical operations that answer the question: "To what power must a certain base be raised to obtain a certain number?" They are the inverses of exponentiation, playing a crucial role in solving equations involving exponential growth or decay, such as probability problems involving repeated events.

• A logarithm \( ext{log}_b(a)\) answers: "What power must \(b\) be raised to get \(a\)?"
• Common logs use base 10, while natural logs use base \(e\) (approximately 2.718)
• If you face an equation like \(b^n = a\), taking the log of both sides rephrases it to \(n \cdot \text{log}_b(b) = \text{log}_b(a)\)

In our problem, we use logarithms to solve \(0.6^n = 0.01\), translating to \(n = \frac{\text{log}(0.01)}{\text{log}(0.6)}\). This lets us find \(n\), the minimum inquiries needed to ensure a 99% chance of renting a room. Using logs in probability helps tackle tricky exponentials with ease.