Random Sleep Times

If Ephram goes to sleep at 10pm he can wake between 6 am and 8 am. Likewise, If he goes to sleep at 11 pm he may wake between 7 am and 9 am. Any other sleep/awake state is enclosed in the parallelogram formed by the vertices ( 6, 10 ),( 8, 10 ),( 7, 11 ),( 9, 11 ) as shown in the graph below:

The probability of Ephram waking after 8:30 is the area of the purple triangle divided by the total area of the parallelogram.

$\begin{aligned} P ( W > 8:30 ) &= \frac{ \frac{1}{2} \cdot 0.5 \cdot 0.5 }{ 2 \cdot 1} \\ &= \left( \frac{1}{2} \right)^4 \end{aligned}$

Thus the probability of him waking before 8:30:

$\begin{aligned} P( W < 8:30 ) &= 1 - P ( W > 8:30 ) \\ &= 1 - \left( \frac{1}{2} \right)^4 \\ &= \frac{15}{16} \end{aligned}$

Thus;

$m+n = 31$