Answer

Suppose R arrives at \(x\) hours. J has to arrive between \(x\) hrs to \(x\) hrs + 15 mins. Similarly if J arrives at \(y\) hours, R has to arrive between \(y\) hours to \(y\) hours + 15 mins. These are regions enclosed by the regions \(x \leq 1, y \leq 1, y \leq x + \frac{1}{4} \;and\; y \geq x - \frac{1}{4}\). The probability is then \(1 - P(\text{not meeting}) = 1 - 2(\frac{1}{2} \frac{3}{4} \frac{3}{4}) = \frac{7}{16}\).