Triangle Probablity Distribution

Question

Given the following probability distribution

Find $P(X<-0.75)+P(X>0.75)$.

I've found that $$P(X<-0.75)+P(X>0.75)=2(\frac{(1-0.75)\cdot h}{2}),$$ where $h$ is the height of the shaded triangle.

How do we get the height?

Answer is $\frac{1}{16}$

Edit 1:

Since this is a probability distribution, the entire area $= 1$.

Area of entire triangle: $\frac{(1-(-1))\cdot H}{2} = 1$

$H = 1$

We split the entire triangle in half vertically down the middle at $0$ so we get two triangles which have all 3 corner angles equal to its respective shaded triangles contained within. Now we use the property of similar triangles where the side-lengths of the shaded triangles are proportionate to its parent triangle.

$h = \frac{1-0.75}{1-0}\cdot H$

$h = \frac{1}{4}\cdot 1$

$h = \frac{1}{4}$

Thus,

$P(X<-0.75)+P(X>0.75)$

$=2(\frac{(1-0.75)\cdot \frac{1}{4}}{2})$

$=\frac{1}{16}$

Answer 1

A probability distribution integrates to $1$, i.e. the area of the big triangle must be $1$. What must its height then be? Now use similar triangles to compute the height of the smaller triangles.

Answer 2

Note that from $-1$ to $0$ you capture $0.5$ of the total probability, then use facts about similar triangles to find the height ($h$) you require.

More specifically, you know that $Pr(-1<X<0) = 0.5$. Hopefully from here on in you can use the expression for the area of a triangle to calculate the height of the triangle as a whole.

Next, use the similar triangle ratio expression to find your required height.

Answer 3

You do not need to find the height. If you join the two shaded small triangles, you will get a triangle similar to large one. Using the property of similar triangles: $$\frac{S_1}{S}=\left(\frac{a_1}{a}\right)^2 \Rightarrow \frac{S_1}{1}=\left(\frac{0.5}{2}\right)^2 \Rightarrow S_1=\frac{1}{16}.$$