- Pick a random integer from 1 to 10,000. What's the probability that every digit of that number is different?
- Roll a die until you roll something greater than 4. What's the expected value of the sum of all the rolls?
- Draw 3 cards from a standard 52-card deck. What's the expected value of the product of the three numbers (J = 11, Q = 12, K = 13)?
- Toss 100 coins. Pair up an H with a T and remove that pair, repeating until you can't pair anymore. What's the expected number of coins left over at the end?
- Roll a die until you get two consecutive rolls that are the same number. What's the expected value of the sum of all the rolls?
- Toss 8 coins. What's the probability of getting three in a row that are the same (three consecutive heads or three consecutive tails)?
- A random walk starting from the origin: each step you roll a die. If it comes up 1, 2, or 3, you move right that many steps. If it comes up 4, 5, or 6, you move left (that number minus 3) steps. On average, how many rounds does it take to get at least 10 steps away from the origin?
- Randomly pick two three-digit numbers. What's the probability that their difference is a two-digit number?
- Two particles sit on opposite vertices of an octagon. Every second, you flip a coin for each particle to decide whether it moves clockwise or counterclockwise by one vertex. On average, how many steps does it take for the two particles to land on the same vertex?
- Keep rolling a die and track a separate running total for each face value (e.g., three rolls of 3 sum to 9 in that face's bucket, three rolls of 1 sum to 3 in that face's bucket). Stop as soon as any single face's running total reaches 100. What's the expected number of even numbers rolled by the time you stop?
- Draw 13 cards from a 52-card deck. What's the probability that none of them is an ace?
- A particle starts at the exact center of a 10x10 grid. Every second you flip two coins: HH moves it one step north, TT moves it one step south, HT moves it one step west, TH moves it one step east. On average, how many seconds does it take for the particle to reach the edge of the grid?
- Roll 3 dice. What's the expected value of 2^(sum of the 3 dice)?
- Roll a die until you get two consecutive rolls that are the same number. What's the expected value of the product of all the rolls?
- You start with $10. Roll a die 10 times: if the roll is even, add to your total; if it's odd, subtract from your total. What's the probability that you go broke at some point during the game?
- You and a friend each randomly pick three numbers from 1 to 10 (inclusive, repeats allowed). What's the probability that you and your friend end up with no numbers in common?
- Keep rolling a die until the running total exceeds 100. What's the probability that the very last die you rolled showed a 2?
Optiver Data Scientist Interview Experience — A Marathon of 17 Probability OA Questions
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