Let \(X\) be a random variable uniformly distributed in the interval [1, 3] and \(Y\) be a random variable uniformly distributed in the interval [2, 4]. If \(X\) and \(Y\) are independent of each other, the probability \(P(X \ge Y)\) is ______ (rounded off to three decimal places).
Answer
0.13
Explanation
By independence, \((X, Y)\) is uniform on the rectangle \([1, 3] \times [2, 4]\), whose area is 4, so a probability is the favourable area divided by 4.
Since \(Y\) is never below 2, \(X \ge Y\) can only happen when \(X \ge 2\). The favourable region is the triangle \(2 \le Y \le X \le 3\), with both legs of length 1:
\[ \text{area} = \int_{2}^{3} (x - 2)\,dx = \frac{1}{2}, \qquad P(X \ge Y) = \frac{1/2}{4} = \frac{1}{8} = 0.125 \]
Answer: 0.125