Solution
One way to look at this is to consider the start and end times. 00:00 - 00:15, 01:00 - 01:15 and 02:00 - 02:15 are the three intervals where both hands are in the region. Note that 03:00 is another instantaneous instance, but this does not affect the proportion.
The total time is 45 minutes every 12 hours. That's \(\frac{1}{16}\) of the day.
Another way of looking at this is that there is \(\frac{1}{4}\) probability of each hand being in the region. So there is \(\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}\) probability of both hands being in the region.
Here's a more challenging one!
Just like before, here are the times where the minute hand is in the upper region and the hour hand is in the other region. 04:00 - 04:15, 05:00 - 05:15, 06:00 - 06:15, 07:00 - 07:15. That's 1 hour in total so far. Note that 08:00 is another instantaneous case.
And if the hands are in the other regions. 00:20 - 00:40, 01:20 - 01:40, 02:20 - 02:40. That's another hour.
In total, that's 2 hours every 12 hour period, which is \(\frac{1}{6}\) of the day.
Again, thinking about probabilities, we could say that the probability of the minute being in the upper region is \(\frac{3}{12}\) and the probability of the hour hand being in the lower region is \(\frac{4}{12}\) which when multiplied together gives \(\frac{12}{144}=\frac{1}{12}\). The probability of the minute and hour hands being in the lower and upper regions, respectively, is the same. Doubling the probability gives \(\frac{1}{6}\) as before.