Probability & Game Puzzles

Probability puzzles (Monty Hall, birthday paradox, coin/dice games, fair-coin-from-biased) test whether you reason with probability correctly rather than trusting intuition — which is famously wrong here. Interviewers love watching candidates fight their gut and compute properly.

Famous results to know

Puzzle	Answer / insight
Monty Hall (switch?)	Always switch → 2/3 win vs 1/3
Birthday paradox	Just 23 people → >50% share a birthday
Fair result from a biased coin	Flip twice; HT vs TH are equally likely (von Neumann)
Expected value of a game	Sum(outcome × probability); play if EV > cost

Exam Tricks & Tips

• 🎯 Compute, don't trust intuition — Monty Hall and the birthday paradox both defy gut feeling.
• 🎯 Monty Hall: always switch — the host's reveal concentrates 2/3 probability on the other door.
• 🎯 Use expected value for game puzzles — EV = Σ(outcome × probability); play only if EV beats the cost.
• 🎯 Get a fair coin from a biased one via von Neumann: flip pairs, HT = heads, TH = tails, discard HH/TT.
• 🎯 Enumerate the sample space for small problems — list outcomes to avoid double-counting.
• ❌ Common mistake: assuming "two options means 50/50" (Monty Hall) — conditional probability, not symmetry, rules.

Expected pattern: Quant/product interviews; a probability brain-teaser you must reason through, resisting the intuitive-but-wrong answer.

Quick recap: Monty Hall → switch (2/3). Birthday → 23 people. Games → expected value. Biased→fair via von Neumann. Compute, don't guess.

Cover the answer, recall, then check. 10 cards on probability & game puzzles.

Q1. Monty Hall — switch or stay?
A1. Always switch — switching wins 2/3 of the time vs 1/3 for staying.

Q2. Why does switching win in Monty Hall?
A2. The host's reveal of a losing door concentrates 2/3 probability on the other unopened door.

Q3. Birthday paradox — how many people for >50% match?
A3. Just 23 people.

Q4. How do you evaluate a game/bet puzzle?
A4. Expected value = Σ(outcome × probability); play only if EV exceeds the cost.

Q5. Get a fair result from a biased coin?
A5. Von Neumann: flip in pairs — HT = heads, TH = tails, discard HH and TT.

Q6. Why are HT and TH equally likely with a biased coin?
A6. Each has probability p(1−p), independent of the bias p.

Q7. Biggest probability-puzzle trap?
A7. Assuming "two options means 50/50" — conditional probability, not symmetry, governs.

Q8. How to handle a small probability problem?
A8. Enumerate the sample space (list outcomes) to avoid double-counting.

Q9. What do probability puzzles really test?
A9. Whether you compute correctly instead of trusting (often wrong) intuition.

Q10. Expected value formula in words?
A10. Sum of each outcome multiplied by its probability.

Probability puzzles — the Monty Hall problem, birthday paradox, fair-coin-from-biased, expected-value games — appear in quant, product, and data-role interviews. They're notorious because intuition is often wrong, so interviewers use them to see whether you reason carefully with the actual math instead of trusting a gut feeling. A few principles and the famous puzzles cover most of what's asked.

What this covers / why it matters: the core techniques for probability and game puzzles (conditional probability, expected value, symmetry) with worked classics. Intuition-defying, so method matters most.

The approach

Beginner — the foundational tools

• Basic probability: P(event) = favourable outcomes / total equally-likely outcomes.
• Conditional probability: P(A given B) — update the sample space once you know B happened. This is where intuition most often fails.
• Expected value: for a game, E = sum of (each outcome's value x its probability). It's the key to "should you play?" and "how many tries on average?" questions.
• Complement trick: P(at least one) = 1 - P(none) — often far easier than summing cases.

Intermediate — the classic puzzles

• Monty Hall: 3 doors, one prize; you pick one, the host opens a different door showing a goat, then asks if you want to switch. You should switch — it wins 2/3 of the time. Your first pick was right 1/3 of the time; the host's action concentrates the remaining 2/3 onto the other unopened door.
• Birthday paradox: in a room of just 23 people, there's a >50% chance two share a birthday — because you compare all pairs (253 of them), not each person to you. Compute via the complement: 1 - P(all distinct).
• Fair result from a biased coin (von Neumann): flip twice; HT = call "heads", TH = call "tails", HH/TT = reflip. Since P(HT) = P(TH) = p(1-p), the result is fair regardless of the bias.

Advanced — expected value and symmetry

For game/dice puzzles, set up the expected value and often a recursive equation. Example: expected number of fair-coin flips to get the first head is E = 1 + (1/2)E (with probability 1/2 you're done, else you've used one flip and start over), giving E = 2. Use symmetry to shortcut: in "two people draw cards, who's more likely to draw the ace first?", symmetry often makes the answer 1/2 or evenly split without heavy computation.

Worked example

Monty Hall, reasoned properly:
"There are 3 doors; I pick door 1. The probability the prize is behind my door is 1/3, and the probability it's behind one of the other two is 2/3. The host — who knows where the prize is — then opens one of the other doors to reveal a goat. Crucially, that doesn't change the 1/3 for my door, but it collapses the entire 2/3 onto the single remaining unopened door, because the host will never open the prize door. So switching wins with probability 2/3, versus 1/3 for staying. I should switch. The intuition trap is thinking it becomes 50/50 after one door opens — but the host's choice is informed, which is what breaks the symmetry."

Expected flips to first head: "Let E be the expected flips. With probability 1/2 I get heads immediately (1 flip); with probability 1/2 I get tails (1 flip) and I'm back where I started, expecting E more. So E = 1 + (1/2)E, which gives E = 2 flips on average."

What interviewers look for

• Careful use of the sample space and conditional probability, not gut feel.
• Expected-value setup for game/repetition puzzles.
• Recognising the classics (Monty Hall, birthday) and why the intuition fails.
• The complement and symmetry shortcuts where they simplify.

Do's and don'ts

• Do write the sample space and use conditional probability explicitly.
• Do use expected value for "how many tries / should I play" questions, and the complement for "at least one".
• Don't trust first-instinct probabilities — verify with the math.
• Don't forget that an informed action (Monty's host) carries information.
• Mnemonic: SPACE — Sample space first, Probability = fav/total, condition on info, Averages via expected value, Complement for 'at least one'.

Worked Example

Problem/Question: The classic Monty Hall puzzle: three doors, a car behind one and goats behind the other two. You pick a door. The host, who knows what's behind them, opens a different door revealing a goat, then asks if you want to switch. Should you?
Solution/Model answer: Yes — switching doubles your winning probability from 1/3 to 2/3.
Reasoning:

• Your initial pick is correct with probability 1/3; the car is behind one of the other two doors with probability 2/3.
• The host knowingly opens a goat door among the other two. This does not change your original 1/3 — but it concentrates the entire 2/3 onto the single remaining unopened door.
• So switching wins whenever your first pick was wrong, which is 2/3 of the time; staying wins only if your first pick was right (1/3).
  Enumerate to confirm: over three equally likely car positions, "always switch" wins in 2 of 3 cases. The counter-intuitive part is that the host's knowledge injects information.
  Answer/Takeaway: Always switch: your initial pick wins 1/3 of the time, so the other door wins 2/3. The host's informed reveal transfers the collective 2/3 probability onto the one remaining door. Enumerating cases (or reasoning about the host's knowledge) resolves the intuition.