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* A line of 100 passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. (For convenience, let's say that the nth passenger in line has a ticket for the seat number n.) Unfortunately, the first person in line is crazy, and will ignore the seat number on their ticket, picking a random seat to occupy. All of the other passengers are quite normal, and will go to their proper seat unless it is already occupied. If it is occupied, they will then find a free seat to sit in, at random. What is the probability that the last (100th) person to board the plane will sit in their proper seat (#100)?
* What is the sum of the numbers one to 100?
* Two entangled quantum prisoners choose Cooperate or Defect in superposition. If measured in a Bell basis, payoffs are superposed. Derive the Nash equilibrium when payoffs are \( \sqrt{2} \$ \) for mutual cooperation.
* You have a 3 gallon jug and 5 gallon jug, how do you measure out exactly 4 gallons? Is this possible?
* You have a 3 gallon jug and 5 gallon jug. How do you measure out exactly 4 gallons? Is this possible?
* You have 17 coins and I have 16 coins, we flip all coins at the same time. If you have more heads then you win, if we have the same number of heads or if you have less then I win. What's your probability of winning?
* What is the probability you draw two cards of the same color from a standard 52-card deck? You are drawing without replacement.
* You're in a room with three light switches, each of which controls one of three light bulbs in the next room. You need to determine which switch controls which bulb. All lights are off to begin, and you can't see into one room from the other. You can inspect the other room only once. How can you find out which switches are connected to which bulbs? Is this possible?