Solve probability and game-theory puzzles

You are interviewing for a **quant trading internship**. Work through the following probability and game-theory puzzles. There's no need to answer instantly — taking a minute or two to reason or compute before committing to an answer is expected. For each puzzle, state your strategy/reasoning clearly, then give the requested numerical result where one is asked for. --- ## 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 exactly **100 rounds**. In each round you choose **exactly one** of two actions: - **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 that bottle is non-empty, you remove **one** ball and you observe that the removal succeeded. If that bottle is empty, you remove nothing. The only feedback you ever get is **whether a removal attempt succeeded** — you never see which bottle was acted on, nor the contents of either bottle. **Goal:** maximize the **expected number of balls removed** over the 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 \(p\), with prior \(p \sim \mathrm{Unif}(0,1)\). **Before each flip**, you must guess Heads or Tails. You earn **\$1** for each correct guess (and \$0 for an incorrect one). You see the outcome of every flip after guessing it, so your guesses may depend on the flips seen so far. **Tasks:** 1. What guessing strategy maximizes your expected total winnings? 2. Let \(X\) be the price to play, paid upfront. Treating yourself as **risk-neutral**, for what values of \(X\) 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 keep a running profit \(S\), starting at \(S = 0\). On each roll: - Roll **2–10**: that value is added to \(S\). - Roll **1**: the game **ends immediately** and your final payout is **\$0** (your accumulated \(S\) is wiped out). **Before each roll** (while the game is still going), you may instead choose to **stop** and take your current \(S\) as your payout. **Tasks — single-player:** Determine the optimal stopping rule. **Tasks — two-player competition:** Two players each play this same game with their **own independent die sequence**, and each chooses when to stop. The player with the **higher final payout** is paid **their own** final payout by a third party; the loser receives **\$0**. (If the payouts tie, they split equally.) Describe how optimal play changes relative to 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 alternate turns in an **ascending-bid** game about the total sum \(a + b\): - On your turn, you 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 \(a + b \ge k\). - When a player challenges, both dice are revealed: - If \(a + b \ge k\), the **last bidder** (the one who made bid \(k\)) wins. - Otherwise, the **challenger** wins. **Task:** Describe an optimal strategy — how to decide whether to raise or challenge, and what value to raise to — as a function of your observed die value \(a\) and the bid history.