Calculating Probability in a Uniform Distribution from -1 to 1

A random number generator outputs values uniformly between -1 and 1. In such a uniform distribution, the probability of selecting a number within a specific interval is directly proportional to the length of that interval.

The Problem:

We are asked to find the probability that a randomly generated number lies between 0.9 and 1.

Understanding Uniform Distribution:

A uniform distribution over an interval $[a,b]$ means:

• Every number between $a$ and $b$ has equal probability of being selected.

• The total probability across the interval is 1.

• The probability density is:

  f(x)=1b−af(x) = \frac{1}{b – a}f(x)=b−a1​

For the interval $[-1,1]$:

• $a=-1$, $b=1$

• Width of the interval = $1-(-1)=2$

• So, the probability density is:

  f(x)=12f(x) = \frac{1}{2}f(x)=21​

Calculating the Desired Probability:

We want the probability that the number lies between 0.9 and 1:

P(0.9≤x≤1)=(1−0.9)×12=0.1×12=0.05P(0.9 \leq x \leq 1) = (1 – 0.9) \times \frac{1}{2} = 0.1 \times \frac{1}{2} = 0.05P(0.9≤x≤1)=(1−0.9)×21​=0.1×21​=0.05

✅ Correct Answer:

(2) 0.05