SIG

Please add rice (points/credits)~ Mid-August: Took the online assessment (OA), received an interview invitation in early September, can choose within 10 business days Late September: Recruiter screening I learned the interview process is: first round HR screening, second round math and probabilities, third round writing a take-home assignment, then discussing the code with an interviewer, and finally flying to Philadelphia for Superday Q1: Three independent random variables x₁, x₂, x₃ are drawn from a uniform distribution on [0,3]. What is the probability that their median lies in [1,2]? Answer: 13/27 Q2: You have 5 dice showing values from 1-9. Your current roll shows: 2, 2, 4, 6, 7. You may reroll exactly one die. You win if you achieve either: Three of a kind plus a pair (e.g., 2, 2, 2, 4, 4) Three consecutive numbers plus a pair (e.g., 2, 2, 5, 6, 7) Which die should you reroll? Answer: Reroll the 4 Q3: Continuing from Q2, compare two scenarios where you can reroll exactly one die: Scenario A: 2, 2, 2, 4, 6 Scenario B: 2, 4, 4, 4, 6 Using the same winning conditions as Q2, which scenario is better, and which die should you reroll in each case? Answer: Both scenarios are equally good. In case A, you can reroll the 6 and win when getting either 3 or 4 (probability = 2/9) In case B, you can reroll 6 to win when getting 2 or 3 (probability = 2/9). Rerolling 2 is similar Q4: Continuing from Q3, after rerolling the 6 in both scenarios, you obtain a new value. For which new values (if any) does one scenario become superior to the other? Answer: Scenario A (2, 2, 2, 4, X): Regardless of the new value X, the optimal strategy remains rerolling one of the dice for a 2/9 win probability. Scenario B (2, 4, 4, 4, X): If X = 5, you have 2, 4, 4, 4, 5. Now rerolling the 2 wins if you roll: 3 → gives 3, 4, 5 consecutive + pair of 4's 5 → gives three 4's + pair of 5's 6 → gives 4, 5, 6 consecutive + pair of 4's This yields a 3/9 = 1/3 win probability, making Scenario B superior when the new roll is 5.