Citadel Securities Quantitative Trading Intern interview questions

James goes to the gym every Monday and Tuesday. On every other day of the week, he independently has a 30% chance of going to the gym. On a random day, James is spotted at the gym. What is the probability it is Monday? A sock drawer has 4 black and 4 white socks. James removes four socks from the drawer. What is the probability that he can make two pairs of same-colored socks? Charlie has 6 hamburger patties and 5 veggie burger patties. To make a massive burger, he stacks all the patties randomly into one burger. What is the expected number of adjacent pairs of patties that are made of the same product? X and Y are random real numbers independently and uniformly picked between 0 and 1. What is the probability that the difference between X and Y is between 3/5 and 4/5? A neighborhood has 20 adults and 11 children. The adults on average spend 80 minutes walking a week, with a standard deviation of 10 minutes. The children on average independently spend 50 minutes walking a week, with a standard deviation of 20 minutes. What is the standard deviation of the total number of minutes walked in the neighborhood? A town has yellow and green buses. The yellow buses are late 1/6 of the time while the green buses are late 1/8 of the time. Sophie gets on a bus randomly and is late. If the probability that she rode a green bus is 1/3, what is the ratio of yellow to green buses in the town? There are several bags containing 2, 3, 4, 5, and 6 candies. For each bag, Charlie independently decides to either take or not take the bag, with equal probability. What is the probability that Charlie takes at least half of the total candies available? Every day for one week, Taylor independently decides to have either eggs or yogurt for breakfast with equal probability. What is the probability that there is a stretch of at least five days where they only eat eggs? The random variable A is uniformly picked between 1 and 2. The random variable B is uniformly picked between 1 and A^2. What is the expected value of B? Fatima has 5 cards numbered from 1 to 5. She shuffles the deck and notices that the first two cards differ by 1. What is the probability that the last card is a 2? You have an unfair coin that lands on heads 1/6 of the time, whereas your opponent has an unfair coin that lands on heads 1/3 of the time. You alternate flipping your respective coins until one lands on heads. If you are going first, what is the probability you get heads before your opponent? Four friends randomly and independently choose to go to one of 4 restaurants in a town. What is the probability that all of the friends end up at the same restaurant or that all of them eat alone? Two teams are playing a "first to three wins" tournament: that is, they will play games until one team has won three games. Each game independently has a ½ chance of being won by either team. What is the expected difference in the number of games won by the overall tournament winner and the overall tournament loser? A frog is attempting two successive jumps to get out of a hole. On its first jump, it has a 20% chance of succeeding; for its second jump afterward, it has a 30% chance of succeeding. If the frog fails, it falls and must try anew. What is the total expected number of jumps the frog will take to get out of the hole?

Probability of various dice/card games