Expected Value: all problems

Mark is playing a game at the casino. The game costs $\$100$. There is a $5\%$ chance he will win $\$500$ and a $20\%$ chance he will win $\$250$ and $75\%$ chance he will win $\$0$. What is the expected value for the money he will lose?

A player pays $\$5$ to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)

Awnik repeatedly plays a game that has a probability of winning of $\frac13$. The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?

A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $t - s$?

Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}$.

For each permutation $a_1, a_2, a_3, \dots, a_{10}$ of the integers $1, 2, 3, \dots, 10$, form the sum

The average value of all such sums can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability $p$. Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors. For what value of $p$ is the expected number of valid squares that the bishop can move to (in one move) equal to the expected number of squares that the knight can move to (in one move)?