Probability of Being Assigned a Prime Numbered Room in a Hotel

One of the common questions in probability theory and real-world scenarios is determining the likelihood of being assigned a specific type of room in a hotel. This article will focus on the scenario where a hotel with rooms numbered from 1 to 19 randomly allocates rooms to guests. Specifically, we will calculate the probability that the first guest arrives and is given a room with a prime number.

Identifying Prime Numbers from 1 to 19

Prime numbers are natural numbers greater than 1 that have exactly two positive divisors: 1 and the number itself. The prime numbers in the range from 1 to 19 are 2, 3, 5, 7, 11, 13, 17, and 19. These are the only numbers within this range that meet the criteria for being prime.

Step-by-Step Breakdown of the Calculation

Identify the Prime Numbers between 1 and 19: The prime numbers in this range are 2, 3, 5, 7, 11, 13, 17, and 19. Step 1: List all the prime numbers. Step 2: Count the total number of prime numbers in this range. Count the Total Number of Rooms: There are 19 rooms in the hotel. Calculate the Probability: The probability of the first guest being assigned a prime-numbered room is calculated as the number of prime-numbered rooms divided by the total number of rooms.

Probability (P) Number of prime-numbered rooms / Total number of rooms

Probability (P) 8 / 19

Various Scenarios and Contexts for Prime Numbered Rooms

The probabilities can change based on the number of rooms and whether certain numbers are excluded due to superstitions or other reasons. For example:

Huge Hotel with 2000 Rooms: If a hotel has 2000 rooms, and 8 of them are prime numbers (2, 3, 5, 7, 11, 13, 17, 19), the probability that a guest is assigned a prime-numbered room is 8/2000, or 1/250. Huge Hotel with 2000 Rooms (General Prime Ratio): Assuming the prime ratio in a huge hotel is approximately 6.2%, the probability would be 125/2000, or 1/16. Small Hotel with 19 Rooms (Excluding Room 13): If a small hotel with 19 rooms excludes room 13 due to superstitions, then there are only 7 prime-numbered rooms (2, 3, 5, 7, 11, 13, 17, 19, excluding 13), resulting in a probability of 7/18, or 4/9.

These various scenarios demonstrate the importance of clearly defining the problem's conditions and assumptions when determining probabilities.

Calculation for Random Integer Assignment

Assuming a guest is assigned a random integer between 1 and 19, the probability of being assigned a prime-numbered room is calculated as follows:

Count the Prime Numbers: There are 8 prime numbers between 1 and 19. Count the Total Outcomes: There are 19 possible rooms. Calculate the Probability: The probability is given by the formula: P Number of Prime Numbers / Total Outcomes

Probability (P) 8 / 19 ≈ 0.42

Conclusion

In conclusion, the probability that a guest is first assigned a room with a prime number in a hotel with 19 rooms is 8/19. This outcome is based on the assumption that all room assignments are equally likely. Adjustments in this probability can be made depending on the specific conditions and assumptions of the problem scenario, such as room exclusions due to superstitions or the size of the hotel.

Keywords: prime numbers, hotel room allocation, probability calculation