X and Y are independent uniform random variables on [0,1]. What is the probability that X+Y < 1?

X and Y are independent uniform random variables on [0,1]. What is the probability that X+Y < 1?

Approach: Geometric probability: area of triangle in unit square.

The region X+Y<1 in the unit square is a right triangle with area 0.5. Probability = 0.5.

Follow-up: What is P(X+Y < 1.5)?

Key concepts: joint distribution, area, uniform.