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Optiver Beat The Odds: Five Rapid-Fire Probability Brainteasers (Dice, Cards, Coins, Pigeonhole)

OptiverSoftware Engineer · Online Assessment
MediumSoftware EngineerOnline AssessmentSoftware Engineering Fundamentals

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This collection of rapid-fire probability brainteasers evaluates probabilistic reasoning, combinatorics, symmetry-based thinking, conditional probability, and fast mental arithmetic to gauge quantitative intuition under time pressure.

You are taking a timed online quantitative assessment (in the style of Optiver's "Beat The Odds" test). It consists of rapid-fire probability questions that must be answered quickly with mental math — no calculator. For each question below, give the exact probability (as a fraction or decimal) and be ready to justify it in one or two sentences.

Constraints & Assumptions

  • All dice are fair six-sided dice; all throws are independent.
  • All coins are fair; flips are independent.
  • The deck is a standard 52-card deck (26 red, 26 black), shuffled uniformly at random.
  • In the real assessment each question allows only a few seconds, so favor the fastest correct reasoning (symmetry, complements, counting) over brute-force enumeration.
  • Answers should be exact (e.g., $5/6$), not rough estimates.

Clarifying Questions to Ask

  • Are the dice and coins fair, and are successive throws/flips independent?
  • Is the deck a standard 52-card deck with 26 red and 26 black cards, and is the shuffle uniformly random?
  • When cards are "thrown in the bin," are they discarded unseen (i.e., we learn nothing about their colors)?
  • Does "different face value" mean strictly different from the first throw's value?
  • Should answers be exact fractions, or rounded decimals to a given precision?

Part 1

You throw one die two times. What is the probability that the second throw shows a different face value than the first throw?

Part 2

Take a shuffled deck of 52 playing cards and throw the top 10 cards in the bin (unseen). What is the probability that the new top card — the 11th card of the original deck — is red?

Part 3

You throw two dice. What is the probability that the two dice add up to 11 or 12?

Part 4

You flip a coin 3 times. What is the probability that the outcome is the same for all flips (all Heads or all Tails)?

Part 5

61 coins are randomly put into 15 boxes. If any box ends up containing more than 4 coins, you win a prize. What is the probability that you win a prize?

Hints

Hint 1

Condition on the first throw: Whatever the first throw shows, the second throw is independent and uniform over 6 faces. How many of those 6 faces "collide" with the first result?

Hint 2

Symmetry: In a uniformly shuffled deck, does any fixed position (1st, 11th, 37th, ...) have a different color distribution than any other?

Hint 3

What information did you gain?: The 10 discarded cards are never observed. If you learn nothing about them, can discarding them change your probability for the next card?

Hint 4

Count outcomes: There are $36$ equally likely ordered outcomes for two dice. List the ordered pairs that sum to 11, and those that sum to 12.

Hint 5

Count sequences: There are $2^3$ equally likely sequences. How many of them are constant? Alternatively: after the first flip, what must each subsequent flip do?

Hint 6

Assume the opposite: Suppose no box contains more than 4 coins. What is the maximum total number of coins the 15 boxes could hold?

Hint 7

Not really a probability question: This is a pigeonhole-principle question wearing a probability costume. Does the random placement mechanism matter at all?

Follow-up questions

  • In Part 2, how does the answer change if the 10 discarded cards are turned face up and you see that 7 of them are red?
  • In Part 5, suppose only 60 coins are placed uniformly at random into the 15 boxes. Is the probability of winning still 1? How would you reason about it?
  • In Part 1, what is the expected number of throws until you first see a face value that repeats an earlier one?
  • In Part 3, generalize: for two fair dice, which target sum $s$ maximizes $P(\text{sum} = s)$, and why?