Probability Analysis: Sandra’s Room Assignment

In a scenario where 16 students must be placed in different rooms, we have rooms with varying numbers of beds: 6 beds, 5 beds, 3 beds, and 2 beds. This article delves into the probability that Sandra will be assigned to the room with 6 beds, applying both basic and conditional probability concepts.

Problem Overview

There are a total of 16 students, and the rooms available contain the following number of beds: 6 beds, 5 beds, 3 beds, and 2 beds.

Basic Probability Calculation

Let's start with the basic probability calculation:

Out of the 16 beds, 6 of them are in the room with 6 beds. The probability that Sandra will be assigned to this room is calculated as follows:

Probability (Sandra being assigned to a room with 6 beds) Number of beds in the room with 6 beds / Total number of beds

Therefore, the probability is:

6 / 16 3 / 8 or 0.375

Conditional Probability

To further explore this, let's introduce the concept of conditional probability.

1. Event A: Sandra is assigned to a room.

P(A) 1 / 16 (since there are 16 students and each has an equal chance of being assigned a room)

2. Event B: The room with 6 beds is selected.

P(B) 6 / 16 (since there are 6 beds in the room with 6 beds out of a total of 16 beds)

Note that the selection of the room depends on the number of beds in that room.

3. Joint Probability P(AB): Sandra is assigned to the room with 6 beds.

P(AB) P(A) * P(B) (1 / 16) * (6 / 16) 6 / 256 3 / 128

Here, events A and B are independent since the probability of Sandra being assigned to a room with 6 beds is not influenced by the selection of a specific room.

Summary and Implications

By analyzing the problem using both basic and conditional probabilities, we can see that the probability of Sandra being assigned to the room with 6 beds is:

6 / 16 3 / 8 or 0.375

This analysis demonstrates the importance of understanding both basic and conditional probability concepts in solving real-world problems, especially in scenarios involving room assignments and bed allocations.

Key Takeaways:

Basic probability: The simple calculation of 6/16. Conditional probability: The joint probability of Sandra being assigned to a room with 6 beds. Independence of events: The room selection process being independent of individual assignments.

Conclusion

Understanding probability is crucial in various applications, from academic settings to more practical scenarios. The analysis of Sandra’s room assignment problem provides a clear example of how to apply basic and conditional probability concepts.