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Compute expectations, conditionals, odds, and distributions

OptiverSoftware Engineer · Technical Screen
MediumSoftware EngineerTechnical ScreenStatistics & Math

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This question evaluates proficiency in fundamental probability and statistics concepts — including expected value and variance, conditional probability, discrete arrival modeling, betting odds and the Kelly criterion, and combinatorics for draws without replacement — and is categorized under Statistics & Math.

Answer fast-probability questions without a calculator:

  1. Expected value and variance: A gamble pays +$10 with probability 0.4 and −$6 otherwise; compute EV and Var.
  2. Conditional probability: Two fair dice are rolled; given that the sum is at least 9, what is the probability at least one die shows 6?
  3. Distribution choice: Arrivals average 3 per minute; model the count in 20 seconds and compute P(X ≥ 2).
  4. Gambling odds: Decimal odds for event A are 2.6 while the true probability is 0.35; compute implied probability, edge/expected return, and the Kelly fraction for a unit bankroll.
  5. Combinatorics without replacement: From a standard 52-card deck, what is the probability of drawing at least one Ace in three draws without replacement? Show quick mental-math steps where possible.