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GATE DA 2024 Q56: P(X ≥ Y) for two independent uniform variables

Question

Let XX be a random variable uniformly distributed in the interval [1,3][1, 3] and YY be a random variable uniformly distributed in the interval [2,4][2, 4]. If XX and YY are independent of each other, the probability P(X≥Y)P(X \ge Y) is ______ (rounded off to three decimal places).

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Numerical answer. No negative marking.
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Answer: 0.125 (official GATE 2024 final answer key)

Concept used: For independent uniforms, probabilities are areas in the rectangle of possible (x,y)(x, y), divided by its total area.

First step: The rectangle [1,3]×[2,4][1,3] \times [2,4] has area 4; x≥yx \ge y only in the triangle inside [2,3]×[2,3][2,3] \times [2,3], of area 12\tfrac{1}{2}.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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