Problem 39 - A First Course in Probability

Question

There are 5 hotels in a certain town. If 3 people check into hotels in a day, what is the probability that they each check into a different hotel? What assumptions are you making?

Step-by-Step Solution

Answer

The probability that all 3 people check into different hotels is \(\frac{24}{25}\). We are assuming that each person has an equal chance of checking into any hotel, and their choices are independent.

Step 1: Find the total number of ways for 3 people to check into hotels

Each of the 3 people can choose from 5 hotels. Since we are considering the choices to be independent, we can find the total number of ways by multiplying the number of choices for each person: 5 * 5 * 5. \(Total\,ways\,to\,check\,in = 5^3\)

Step 2: Find the total number of ways for 3 people to check into different hotels

For the first person, there are 5 hotels to choose from. After the first person chooses a hotel, there will be 4 hotels left for the second person to choose from. And then the third person will have 3 hotels left to choose from. We can multiply these numbers to find the total number of ways for all of them to check into different hotels: \(Total\,ways\,to\,check\,into\,different\,hotels = 5 * 4 * 3\)

Step 3: Calculate the probability of all 3 people checking into different hotels

To find the probability, we need to divide the total number of ways for them to check into different hotels by the total number of ways for them to check into hotels: \(Probability\,of\,checking\,into\,different\,hotels = \frac{5 * 4 * 3}{5^3} = \frac{120}{125}\)

Step 4: Simplify the probability

We can simplify the probability by dividing the numerator and the denominator by the greatest common divisor (5): \(\frac{120}{125} = \frac{24}{25}\) So, the probability that all 3 people check into different hotels is \(\frac{24}{25}\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, combination, and permutation of sets. It's an essential component in calculating probabilities because it helps us determine the number of possible outcomes in a given situation.

For example, in the exercise provided, we are tasked with understanding how 3 people can check into 5 different hotels. We use combinatorics to calculate the total number of ways this can happen. The field of combinatorics encompasses various counting principles, such as the Rule of Product, which we used to determine the total number of ways 3 people can check into any of the 5 hotels. By multiplying the choices for each person, we apply this fundamental principle.

In more complex situations, combinatorics can be involved with various other principles like combinations and permutations, which are methods of counting without and with regard to the order of selection, respectively. Understanding the basics of these principles is crucial in solving a range of probability problems.

Independent Events

Independent events in probability are those whose outcomes do not affect each other. When we say that events are independent, the occurrence of one event does not change the probability of another occurring. This concept is fundamental when calculating the probabilities of multiple events.

In the context of our hotel exercise, the assumption is that each person checks into a hotel independently of one another. There are no constraints like 'a hotel can only accommodate one person per day'. It means that one person choosing a particular hotel does not influence the choices available to the others, at least for the total number of ways to check into hotels.

When dealing with independent events, the total probability is the product of the individual probabilities. However, when events are not independent, other methods such as conditional probability would be used. Recognizing whether events are independent is a critical step in any probability calculation.

Permutations

Permutations refer to the number of ways in which a set of objects can be ordered or arranged. When we are interested in the arrangement or the order of selection, permutations come into play. Mathematically, the number of permutations of 'n' objects taken 'r' at a time is given by the factorial of 'n' divided by the factorial of 'n-r'.

In the hotel exercise, we are interested in permutations because it matters which person goes into which hotel. The first person has 5 options, the second has 4, following the choice of the first person, and the third will have 3, based on the previous selections. The product of these options, 5 * 4 * 3, is the permutation of 5 hotels taken 3 at a time without repetition, demonstrating that the order of how they check-in is significant.

Understanding how permutations work is crucial when solving problems where the order of events or choices matters, such as seating arrangements, rankings, or, as in our case, hotel check-ins.