Discrete Mathematics
# Expected Value

There are 3 kegs of water. The first keg has 1 liter, the second keg has 2 liters, and the third keg has 3 liters of water in it.

Each day our traveler picks one keg at random with each keg having the same chance of being picked, and drinks one liter of water from it. When he empties a keg, he throws it away. When he is left with one keg, he records the amount of water in it.

What is the expected amount of water (in liters) remaining in that last keg?

If this expected value can be expressed as \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, enter \(a+b\).

\(\)

**Note:** You may use a calculator if you want.

**Bonus:** Solve this problem for 5 kegs.

A class of 30 students contains 15 boys and 15 girls. Their classroom has 15 desks, and exactly 2 students can sit at each desk.

At the start of lessons one morning, the 30 students go into their classroom and seat themselves randomly, so that all possible arrangements of pupils are equally likely. The expected number of girls who end up sitting at a desk with another girl can be written as \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

In the digital board game Armello, players battle by rolling dice. Each die has six sides: Sword, Shield, Sun, Moon, Wyld, and Rot. In battles, each player has a specific number of dice, and they roll them all.

- A result of a Sword gives one attack.
- A result of a Shield gives one defense.
- A result of a Sun gives one attack if the turn is a daytime turn, otherwise it doesn't give anything.
- A result of a Moon gives one attack if the turn is a nighttime turn, otherwise it doesn't give anything.
- A result of a Wyld gives one attack, and the die
**explodes**, if the player is not Corrupted; otherwise it doesn't give anything. - A result of a Rot gives one attack, and the die
**explodes**, if the player is Corrupted; otherwise it doesn't give anything.

A turn is either daytime or nighttime (but not both). A player is either Corrupted or not Corrupted (but not both). When a die **explodes**, the player rolls an additional die; this die can explode again, giving more dice to roll, and so on. (The result before the die explodes stands.)

After all dice have settled, the attacks and defenses are tallied up. Each attack is blocked by an opposing defense; the rest of the attack, if any, is dealt as damage. For example, if player A has 4 attack and player B has 2 defense, then A deals 2 damage to B. If player A has 2 attack and player B has 4 defense, then both attacks are successfully blocked; B doesn't get any damage. (And the remaining 2 defense is useless.)

Suppose A and B are battling; both start with one die each. The expected amount of damage dealt by A to B can be represented as \(\frac{p}{q}\), where \(p \) and \(q\) are coprime positive integers. Find \(p+q\).

In my favorite game, I start with 3 lives. In each round, I have a 10% chance of dying, losing a life. If I don't die, I am given an extra life, but I cannot ever have more than 3 lives.

What is the expected value for the number of rounds I will play before losing the game (when I lose my final life)?

**Note:** Each round is counted, even if you lose a life in the round.

A row of street parking has spaces numbered 1 through 100, in that order. Each person trying to park drives by the spaces one-by-one starting at number 1.

At each open space, they are told how many open spaces, \(S,\) they haven't passed (including the current space). They park in that space with probability \(\frac{1}{S};\) otherwise, they move on to the next open space.

What is the expected value for the parking spot of the \(100^\text{th}\) person to arrive (assuming no one has left and only one car goes through the parking process at a time)?

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