Mastering Probability Combinatorics: My Challenging Interview at Old

old | | Interview Experience

Interview Date: Not specified
Result: Not specified
Difficulty: Not specified

Interview Process

The interview consisted of 35 questions, all identical and presented in the same order. The duration of the interview was 25 minutes.

Technical Questions

  1. Probability: What is the probability that the sum of two dice is 8?
  2. Combinatorics: In a group of 26 people, how many total handshakes occur if everyone shakes hands with everyone else?
  3. Probability: A company produces small parts, with 1 in 50 being defective. If each good part earns $3 and each defective part costs $75 in repair, what is the expected profit per part?
  4. Probability: I just rolled a 9 with two dice. What is the probability that one of the dice showed a 5?
  5. Probability: I have 30 unique cards numbered from 1 to 30. If you randomly draw one card, I will pay you the number on the card multiplied by $6. What is the expected value of playing this game?
  6. Probability: Jar A contains 3 red balls and 5 black balls, and Jar B contains 4 red balls and 5 black balls. What is the probability that the two balls drawn from each jar are of the same color?
  7. Probability: I have a jar with 15 balls numbered from 1 to 15. If I draw two balls without replacement, what is the probability that the sum of the numbers on the two balls is odd?
  8. Probability: What is the probability of drawing a pair of Jacks from a deck of cards when drawing the first two cards?
  9. Probability: What is the probability of getting the same side when flipping a coin five times?
  10. Probability: A player throws two coins in the air. If either shows heads, they win $2; if both show tails, they lose $7. What is the fair value of this game?
  11. Probability: A jar contains 12 coins, 10 of which are fair (50% heads, 50% tails), and 2 are biased “double heads” coins. If I randomly select one coin and flip it 5 times, resulting in heads each time, what is the probability that I selected a “double heads” coin?
  12. Probability: I start with $64 and bet half of my money on a coin flip six times. If I win three times and lose three times, how much do I gain or lose?
  13. Probability: An auto insurance company knows that unsafe drivers have a 40% chance of having an accident in a year, while safe drivers have only a 10% chance. If 85% of drivers are safe and the company observes an accident, what is the probability that the driver is unsafe?
  14. Probability: A jar contains 7 red balls and 10 black balls. If four balls are drawn without replacement, what is the probability that the first two are red and the last two are black?
  15. Probability: A closet contains 5 pairs of shoes. What is the probability of randomly selecting 5 shoes such that none form a complete pair?
  16. Probability: What is the probability of drawing a pair from a deck of cards when drawing the first two cards?
  17. Probability: You and a friend play a coin flip game. If the sequence “HH” appears first, you win; if “TH” appears first, your friend wins. What is your probability of winning this game?
  18. Probability: In a horse race with 8 horses, what is the probability that horse number 4 comes in second and horse number 1 finishes in the top three?
  19. Probability: We are flipping a fair coin. If heads, I score a point; if tails, you score a point. We continue flipping until one of us reaches three points. What is the expected score of the loser?
  20. Probability: What is the probability that the sum of three dice is 14?
  21. Combinatorics: A person has 9 friends and will invite 6 to a party. If two friends do not get along and cannot attend together, how many possible combinations of invitations are there?
  22. Combinatorics: How many unique circular bracelets can be made with 3 red beads and 3 green beads?
  23. Probability: What is the probability of being dealt a “Blackjack” (drawing two cards without replacement, one being a 10 to King and the other an Ace)?
  24. Combinatorics: How many palindromic arrangements exist for the word “MISSISSIPPI”?
  25. Probability: A small university offers three majors: Computer Science, Mathematics, and Philosophy. 50% of students major in Mathematics, 60% in Computer Science, and 20% in Philosophy; 6% major in all three. What percentage of students major in exactly two fields?
  26. Probability: An energy company is drilling for oil, with a 35% success rate and a 65% failure rate. If they find oil, the stock is worth $100; if they fail, it’s worth $16. The current stock price is $58.50. You have the right to buy the stock for $70 the day after they learn the outcome. What is the fair value of this right?
  27. Probability: I want to throw a party this weekend but worry about rain (70% chance on Saturday, 60% on Sunday). I will choose to have the party on the first sunny day. Fast forward to Monday, my friends loved the party. What is the probability that the party was on Saturday?
  28. Probability: If two squares are randomly selected from an 8x8 chessboard, what is the probability that the squares share the same row or column?
  29. Probability: I drew two cards without replacement from a standard deck of 52 cards, and they were a pair of 8s. I now plan to draw two more cards. What is the probability that the next two cards are also a pair?
  30. Probability: I have a bag with 3 red balls and 5 black balls. I will draw balls without replacement, one by one. What is the probability of drawing the last red ball by the fifth draw?
  31. Probability: I have a shuffled deck of cards and keep flipping until I reveal the first Ace, which is the 33rd card. What is the probability that the next card is the 7 of Diamonds? (The answer is not 1/51)
  32. Probability: You observe a candy conveyor belt and notice that for every five chocolates, four are followed by a soft-centered candy, and for every three soft-centered candies, two are followed by a chocolate. What fraction of the candies on the conveyor belt are soft-centered candies?
  33. Probability: You and a friend will play a coin flip game. If three heads appear consecutively first, you win; if four tails appear consecutively first, your friend wins. What is your probability of winning this game?
  34. Probability: A box contains 3 large donuts and 9 small donuts. Every hour, Homer randomly selects a donut from the box. If it’s a small donut, he eats it; if it’s a large donut, it’s cut in half, one half is eaten, and the other half is returned as a small donut. How many hours until only small donuts remain in the box?

Tips & Insights

  • Focus on understanding probability and combinatorial concepts thoroughly, as many questions revolve around these topics.
  • Practice solving problems under time constraints to improve speed and accuracy.
  • Familiarize yourself with common probability distributions and their applications.