Geometric Probability: all problems

Alex and Bob decide to meet at the park between 3 and 4 pm. They both will arrive at a random time during the hour but will only wait up to 15 minutes for the other person and then leave the park. What is the probability that they will meet?

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

Sonya the frog chooses a point uniformly at random lying within the square $[0, 6] \times [0, 6]$ in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from $[0, 1]$ and a direction uniformly at random from north, south, east, west. All the choices are independent. She now hops the distance in the chosen direction. What is the probability that she lands outside the square?

Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0, 1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x - y| > \frac12$?

Three real numbers are chosen independently and uniformly at random between 0 and 1. What is the probability that the greatest of these three numbers is greater than 2 times each of the other two numbers? (In other words, if the chosen numbers are $a \ge b \ge c$, then $a > 2b$.)

A point $P$ is chosen at random inside square $ABCD$. The probability that $\overline{AP}$ is neither the shortest nor the longest side of $\triangle APB$ can be written as $\frac{a + b\pi - c\sqrt{d}}{e}$, where $a$, $b$, $c$, $d$, and $e$ are positive integers, $\gcd(a, b, c, e) = 1$, and $d$ is not divisible by the square of a prime. What is $a + b + c + d + e$?

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40\%$, and $m = a - b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c$.

A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m + n$.