Solve probability and game-theory puzzles

You are interviewing for a quant trading internship. Answer the following probability / game-theory puzzles. ## 1) Two bottles, add/remove balls (100 rounds) You have **two bottles** (A and B), initially empty, and an unlimited supply of identical balls. The game lasts **100 rounds**. In each round, you choose exactly one action: - **Add**: One ball is placed into **a uniformly random bottle** (A or B). You do **not** observe which bottle received the ball. - **Remove**: One bottle is chosen **uniformly at random** (A or B). If the chosen bottle is non-empty, you remove **one** ball (and you observe that you removed a ball). If the chosen bottle is empty, you remove nothing. **Goal:** maximize the **expected number of balls removed** over 100 rounds. **Tasks:** 1. Describe an optimal strategy. 2. Compute the expected number of balls removed under that strategy. ## 2) 100 coin flips with unknown bias A coin is flipped **100 times**. The probability of Heads is an unknown parameter \(p\), where \(p \sim \mathrm{Unif}(0,1)\) a priori. Before each flip, you must guess Heads or Tails. You earn **$1** if your guess is correct. **Tasks:** 1. What guessing strategy maximizes expected winnings? 2. Let \(X\) be the price to play (paid upfront). For what values of \(X\) (risk-neutral) would you be willing to play? ## 3) Stop/continue with a 10-sided die (single-player and competitive) A fair 10-sided die (faces 1–10) is rolled repeatedly. You maintain a running profit \(S\), starting at \(S=0\). - If you roll **2–10**, you add that value to \(S\). - If you roll **1**, the game **ends immediately** and your final payout is **0**. Before each roll (while the game has not ended), you may choose to **stop** and take your current \(S\) as payout. **Tasks (single-player):** Determine the optimal stopping rule. **Tasks (two-player competition):** Two players independently play this same game (each with their own die sequence). Each chooses when to stop. The player with the **higher final payout** receives **their** payout from a third party; the loser receives **$0**. (If tied, split equally.) Describe how optimal play changes vs. the single-player case. ## 4) Bidding the sum of two dice with private information Two fair six-sided dice are rolled, one assigned to each player. - You see only **your** die outcome \(a\). - Your opponent sees only **their** die outcome \(b\). Players take turns in an **ascending bid** game about the total sum \(a+b\): - On your turn, either **raise** to a strictly higher integer bid \(k\in\{2,\dots,12\}\), or **challenge** the previous bid. - A bid \(k\) is a claim that the true sum satisfies \(a+b \ge k\). - If a player challenges, the dice are revealed: - If \(a+b \ge k\), the **last bidder** wins. - Otherwise, the **challenger** wins. **Task:** Describe an optimal strategy (how to decide whether to raise or challenge, and what to raise to) as a function of your observed die value and the bid history.