In a Uniform(−1,1) random variable $X$, find $P(−0.5 ≤ X < 1.5 \mid X > − 0.25)$

Question

I solved it like this:

$P(−0.5 \le X \le 1.5 \mid X > −0.25) = \cfrac{P(X > − 0.25 \bigcap −0.5 \le X \le 1.5)}{P(X > −0.25)} = \cfrac{\frac 58}{\frac 58} = 1$

$P(−0.5 ≤ X < 1.5) = \int_{-.25}^1 \frac 12 \mathrm dx + \int_1^{1.5} 0 \text dx = \frac 58$

$P(X > − 0.25) = \int_{-0.25}^1 \frac12 \text dx = \frac 58$

Answer 1

You are right !

$P[X>-0.25]=\frac{5}{8}$

$P[(X>-0.25) \cap (-0.5\le X<1.5)]=P[-0.25\le X<1.5]=\frac{5}{8}$

Hence $P[ -0.5\le X<1.5 | X>-0.25]=\frac{P[(X>-0.25) \cap (-0.5\le X<1.5)]}{P[X>-0.25]}=\frac{5/8}{5/8} =1$