Simple Answer:

The probability that Alice and Bob meet is $\frac{7}{16}$.

Step-by-Step Answer:

1. Define the Problem: The meeting window is from 1:00 PM to 2:00 PM, which is 60 minutes. Both Alice and Bob arrive randomly within this window and wait for 15 minutes.

2. Consider the Time Range: The only way they fail to meet is if the time difference between their arrival times is more than 15 minutes.

3. Visualize on a 60×60 Square:

   • Imagine a square plot where one side represents Alice’s arrival time and the other represents Bob’s. The entire square represents all possible combinations of their arrival times within the hour.
   • The square has an area of $60\times 60=3600$ square units, where each unit represents a unique combination of arrival times (in minutes).

4. Calculate Non-meeting Zones:

   • The non-meeting zones are two triangles where the difference between their arrival times is greater than 15 minutes. Each triangle’s area represents the scenarios where they miss each other.
   • The area of each triangle is $\frac{1}{2}\times 45\times 45=1012.5$ square units. The total non-meeting area is $2\times 1012.5=2025$ square units.

5. Calculate Meeting Probability:

   • Subtract the non-meeting area from the total area to find the meeting area: $3600-2025=1575$ square units.
   • The probability of meeting is the ratio of the meeting area to the total area: $\frac{1575}{3600}=\frac{7}{16}$.

Therefore, the step-by-step calculation also shows that the probability of Alice and Bob meeting is $\frac{7}{16}$, confirming the simple answer provided earlier. This approach combines geometric visualization with basic probability principles to solve the problem.