Maria and Joe are jogging towards each other on a long straight path. Joe is running at $10$ mph and Maria at $8$ mph. When they are $3$ miles apart, a fly begins to fly back and forth between them at a constant rate of $15$ mph, turning around instantaneously whenever it reachers one of the runners. How far, in miles, will the fly have traveled when Joe and Maria pass each other?
1. Determine the combined speed at which Maria and Joe are closing the gap. Joe is running at \(10\) mph and Maria at \(8\) mph. Therefore, their combined speed is:
   \[
   10 \text{ mph} + 8 \text{ mph} = 18 \text{ mph}
   \]

2. Calculate the time it takes for Maria and Joe to meet. They start \(3\) miles apart and are closing the gap at \(18\) mph. The time \(t\) it takes for them to meet is given by:
   \[
   t = \frac{\text{distance}}{\text{combined speed}} = \frac{3 \text{ miles}}{18 \text{ mph}} = \frac{1}{6} \text{ hours}
   \]

3. Determine the distance the fly travels. The fly is flying at a constant speed of \(15\) mph. The time it flies is the same as the time it takes for Maria and Joe to meet, which is \(\frac{1}{6}\) hours. Therefore, the distance \(d\) the fly travels is:
   \[
   d = \text{speed} \times \text{time} = 15 \text{ mph} \times \frac{1}{6} \text{ hours} = \frac{15}{6} \text{ miles} = \frac{5}{2} \text{ miles}
   \]

Conclusion:
\[
\boxed{\frac{5}{2}}
\]