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Was asked this question in an interview, so not sure what the optimal answer is, but here goes:

Given a rectangle with length l and breadth b, I pick 4 random points inside it. What is the probability that the 4 points lie on the same half of the rectangle?

Notes:

1. Two halves of the rectangle are defined by a single line passing through the intersection of its diagonals. So the rectangle cannot be divided into two halves of any arbitrary shapes of equal length.
2. A line passing through a picked point and the center does not contain another picked point. That is, A line passing through any two picked points does not pass through the center of the rectangle (Intersection of the diagonals).

Addition: (Maybe a Hint)

I was asked about 3 points first. They saw my approach and were satisfied and then asked me what if there were 4 points.

Answer

9/16

The one to prove it gets the check mark.

I was asked this question in an interview, so I am not sure what the optimal answer is. But here it goes:

Given a rectangle with length l and breadth b, I pick 4 random points inside it. What is the probability that the 4 points lie in the same half of the rectangle?

Notes:

1. Two halves of the rectangle are defined by a single line passing through the intersection of its diagonals. So the rectangle cannot be divided into two halves of any arbitrary shapes of equal length.
2. A line passing through a picked point and the center does not contain another picked point. That is, A line passing through any two picked points does not pass through the center of the rectangle (= intersection of the diagonals).

Addition: (Maybe a Hint)

I was asked about 3 points first. They saw my approach and were satisfied and then asked me what if there were 4 points.

Answer

9/16

The one to prove it gets the check mark.