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.