In the following figures, what is the probability of picking a point in the shaded area?

Let's solve the problem using two method

Method 1: Logic

Divide the triangle into small triangles of qual are as shown in the figure below



There are 16 triangles now with the same area and 13 of them are shaded

=> Probability of picking a point in the shaded area = \(\frac{Shaded }{ Total}\) = \(\frac{13}{16}\)

Method 2: Algebra

Let side of smaller triangle is 1
=> Side of bigger triangle = 4

=> Probability of picking a point in the shaded area = \(\frac{Area Of Shaded Region}{Area Of Bigger Triangle}\) = ( (\(\frac{\sqrt{3}}{4}*4^2\) - 3*\(\frac{\sqrt{3}}{4}*1^2\)) / \(\frac{\sqrt{3}}{4}*4^2\) ) = \(\frac{16 - 3 }{ 16}\) = \(\frac{13}{16}\)

So, Answer will be \(\frac{13}{16}\)
Hope it helps!