Understanding the Probability of the Sum of Two Random Numbers

Introduction

Probability is a fascinating concept that helps us understand randomness in various real-world scenarios. In this article, we explore the probability that the sum of two randomly selected numbers between 0 and 1 falls within the range of 0.5 to 1.5.

Key Factors in the Calculation

• We generate two independent random numbers X and Y, both uniformly distributed between 0 and 1.
• The sum of these numbers is given by S = X + Y.
• We need to determine the probability that 0.5 ≤ S ≤ 1.5.

Defining the Probability Space

The total possible outcomes form a square in the X-Y plane, with both axes ranging from 0 to 1. The sum S = X + Y creates a triangular probability distribution.

Step-by-Step Calculation

The sum of two uniform random variables between 0 and 1 forms a triangular distribution between 0 and 2. The total area of the square is 1 × 1 = 1.

1. The probability that S < 0.5 is represented by a small triangular area with vertices at (0,0), (0.5,0), and (0,0.5). Its area is: 
2. The probability that S > 1.5 is represented by another small triangular area with vertices at (1,1), (1,0.5), and (0.5,1). Its area is: 
3. The probability of the sum falling outside the range (S < 0.5 or S > 1.5) is: 
4. Therefore, the probability that 0.5 ≤ S ≤ 1.5 is: 

Thus, the correct answer is Option B: 75%.

Conclusion

This problem demonstrates a fundamental concept in probability theory. The sum of two uniformly distributed random variables forms a triangular probability density, making it easy to compute probabilities using simple geometry.

Understanding such concepts can help in various fields, including statistics, finance, and machine learning. Stay tuned for more insightful probability problems!

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Probability of sum of two random numbers between 0 and 1