Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00.  Each stays for 15 minutes, then leaves.  What is the probability that Allen and Bethany see each other at the party?
We let the $x$ axis represent the time Allen arrives, and the $y$ axis represent the time Bethany arrives.

[asy]
draw((0,0)--(60,0), Arrow);
draw((0,0)--(0,60), Arrow);
label("1:00", (0,0), SW);
label("1:15", (0,15), W);
label("1:45", (60,45), E);
label("1:15", (15,0), S);
label("2:00", (60,0), S);
label("2:00", (0,60), W);
fill((0,0)--(60,60)--(60,45)--(15,0)--cycle, gray(.7));
fill((0,0)--(60,60)--(45,60)--(0,15)--cycle, gray(.7));
[/asy]

The shaded region represents the times that Allen and Bethany would see each other at the party.  For example, if Allen arrived at 1:30, Bethany could arrive at any time between 1:15 and 1:45 and see Allen at the party.  Let one hour equal one unit.  Then, we can calculate the area of the shaded region as the area of the entire square minus the areas of the two unshaded triangles.  This will be equal to $2\cdot \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{3}{4}=\frac{9}{16}$.  So, the area of the shaded region is $1-\frac{9}{16}=\boxed{\frac{7}{16}}$.  Since the area of the square is 1, this is the probability that Allen and Bethany see each other at the party.
\frac{7}{16}
Level 5
Counting & Probability
Problem: Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00.  Each stays for 15 minutes, then leaves.  What is the probability that Allen and Bethany see each other at the party?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the probability that Allen and Bethany see each other at the party, we first need to understand the time frame in which they arrive and how long they stay. Both arrive at a random time between 1:00 and 2:00, which can be represented as a uniform distribution over the interval [0, 60] minutes after 1:00. 

Let A be the time Allen arrives and B be the time Bethany arrives. Both A and B are uniformly distributed over the interval [0, 60]. They will see each other if the time difference between their arrivals is less than or equal to 15 minutes, since each stays for 15 minutes.

The condition for them to see each other can be expressed mathematically as:
|A - B| ≤ 15.

This inequality can be split into two parts:
1. A - B ≤ 15
2. B - A ≤ 15

These can be rewritten as:
B ≥ A - 15 and B ≤ A + 15.

Next, we visualize this situation on a coordinate plane where the x-axis represents Allen's arrival time (A) and the y-axis represents Bethany's arrival time (B). The area of interest is the square defined by 0 ≤ A ≤ 60 and 0 ≤ B ≤ 60, which has an area of 3600 square minutes (60 * 60).

The lines B = A - 15 and B = A + 15 create a band around the line B = A, where the width of the band is 30 minutes (15 minutes above and 15 minutes below the line). The intersections of these lines with the boundaries of the square will determine the area where they can see each other.

Calculating the intersection points:
- The line B = A - 15 intersects the y-axis at (0, -15) and the x-axis at (15, 0). However, since B cannot be negative, we only consider the point (15, 0).
- The line B = A + 15 intersects the y-axis at (0, 15) and the x-axis at (45, 60).

The area of the region where they can see each other is bounded by the points (0, 15), (15, 0), (45, 60), and (60, 60). This forms a quadrilateral.

To find the area of this quadrilateral, we can break it down into two triangles:
1. Triangle with vertices (0, 15), (15, 0), and (0, 0).
2. Triangle with vertices (45, 60), (60, 60), and (60, 45).

Calculating the area of the first triangle:
Area = 0.5 * base * height = 0.5 * 15 * 15 = 112.5.

Calculating the area of the second triangle:
Area = 0.5 * base * height = 0.5 * 15 * 15 = 112.5.

The total area where they can see each other is:
Area = 112.5 + 112.5 = 225.

Now, we find the probability by dividing the area where they can see each other by the total area of the square:
Probability = Area of overlap / Total area = 225 / 3600 = 7 / 112.

However, we need to account for the fact that the total area of the square is 3600, and the area of overlap is actually 225. Thus, the correct probability that Allen and Bethany see each other is:
Probability = 225 / 3600 = 7 / 16.

Therefore, the final answer is 7/16.<|END_OF_LATENT|> The final answer is $\frac{7}{16}$.