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Two independent random variables \(X\) and \(Y\) are uniformly distributed in the interval \([-1,1]\). The probability that \(\max(X,Y)<\tfrac12\) is

1. 3/4
2. 9/16
3. 1/4
4. 2/3

Correct answer: 9/16

Solution

Since \(X\) and \(Y\) are independent, \(P(\max(X,Y)<1/2)=P(X<1/2, Y<1/2)=P(X<1/2)P(Y<1/2)\). For a uniform variable on \([-1,1]\), \(P(X<1/2)=\frac{1/2-(-1)}{2}=\frac34\), so the required probability is \((3/4)^2=9/16\).

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