Question

Consider a uniform distribution created by a random number generator. The distribution looks like a square with a length of 1 and a height of 1. The random number generator creates any number between 0 and 1. Find the following probabilities:
a) P(0 <= X <= 0.4)
b) P(0.4 <= X <= 1)
c) P(X > 0.6)
d) P(X <= 0.6)
e) P(0.23 <= X <= 0.76)

Answer 1

The rigorous way to do this would be to compute a few integrals, but less work is usually better. Each of these probabilities correspond to the areas of rectangles. Each rectangle will have a height of 1, and the probability of interest will tell you everything you need to know about their lengths.

$\mathbb P(0\le X\le0.4)\implies \text{length}=0.4\implies\mathbb P(0\le X\le0.4)=0.4\times1=0.4$

$\mathbb P(0.4\le X\le1)\implies \text{length}=0.6\implies\mathbb P(0\le X\le0.4)=0.6$

Alternatively, you can use the fact that $\mathbb P(0.4\le X\le 1)=1-\mathbb P(0\le X\le 0.4)$ to get the same result.

$\mathbb P(X>0.6)=\mathbb P(0.60.6)=0.4$

$\mathbb P(X\le0.6)\implies \text{length}=0.6\implies\mathbb P(X\le0.6)=0.6$

$\mathbb P(0.23\le X\le0.76)\implies \text{length}=0.76-0.23=0.53\implies\mathbb P(0.23\le X\le0.76)=0.53$