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 \(\frac{24}{25}\).