1) 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? 2) 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? 3) 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? 4) 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? 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? 6) 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? 7) 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? 8) 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? 9) 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? 1 0) 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 the last card is a 2? 1 1) 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? 1 2) 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? 1 3) 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 1/2 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? 1 4) 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?

This Statistics & Math mini-set evaluates probabilistic reasoning and expectation computation, covering combinatorics, conditional probability, discrete and continuous uniform distributions, variance aggregation, and stochastic-process intuition relevant to data-science quantitative skills.

How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Take-home Project rounds at Citadel.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Citadel during technical interviews.

Solve probability and expectation problems | Citadel Interview Question

Probability and Statistics Mini-Set

Context: Answer each item independently. Unless otherwise specified, assume independence and uniform randomness; days of the week are equally likely.

James goes to the gym every Monday and Tuesday. On every other day (Wednesday–Sunday), he independently has a 30% chance of going. 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 4 socks uniformly at random (without replacement). What is the probability he can make two pairs of same-colored socks?
Charlie has 6 hamburger patties and 5 veggie patties. He stacks all 11 patties in a uniformly random order. What is the expected number of adjacent pairs of patties that are of the same type?
X and Y are independent Uniform(0, 1) random variables. What is the probability that the absolute difference |X − Y| is between 3/5 and 4/5?
A neighborhood has 20 adults and 11 children. Adults spend on average 80 minutes walking per week with standard deviation 10 minutes. Children spend on average 50 minutes per week with standard deviation 20 minutes. Assuming independence across people, what is the standard deviation of the total weekly minutes walked in the neighborhood?
A town has yellow and green buses. Yellow buses are late 1/6 of the time; green buses are late 1/8 of the time. Sophie boards a bus uniformly at random and is late. If the probability she rode a green bus given she was late is 1/3, what is the ratio of yellow to green buses in the town?
There are bags containing 2, 3, 4, 5, and 6 candies (one bag of each). Charlie independently takes each bag with probability 1/2. What is the probability Charlie takes at least half of all candies available?
Each day for one week (7 days), Taylor independently chooses eggs or yogurt for breakfast with equal probability. What is the probability there is a stretch of at least 5 consecutive days where Taylor only eats eggs?
A ~ Uniform[1, 2]. Conditional on A = a, B ~ Uniform[1, a^2]. What is E[B]?
Fatima has cards numbered 1 to 5, shuffled uniformly at random. She observes the first two cards differ by 1. What is the probability the last card is a 2?
You have an unfair coin with P(H) = 1/6; your opponent’s coin has P(H) = 1/3. You alternate flipping your respective coins until the first heads appears. You go first. What is the probability you get heads before your opponent?
Four friends independently and uniformly choose one of 4 restaurants. What is the probability that either all four end up at the same restaurant or that all four eat alone (i.e., all choose distinct restaurants)?
Two teams play a “first to three wins” series (best-of-5). Each game is fair (P(win)=1/2 for each team), independent across games. What is the expected difference in the final win counts between the tournament winner and loser?
A frog attempts two successive jumps to escape a hole: the first jump succeeds with probability 0.2; if that fails, a second jump succeeds with probability 0.3. If it fails the second jump, it falls and must try anew (starting again with the first jump). What is the expected total number of jumps until it escapes?

Probability and Statistics Mini-Set

Context: Answer each item independently. Unless otherwise specified, assume independence and uniform randomness; days of the week are equally likely.

James goes to the gym every Monday and Tuesday. On every other day (Wednesday–Sunday), he independently has a 30% chance of going. 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 4 socks uniformly at random (without replacement). What is the probability he can make two pairs of same-colored socks?
Charlie has 6 hamburger patties and 5 veggie patties. He stacks all 11 patties in a uniformly random order. What is the expected number of adjacent pairs of patties that are of the same type?
X and Y are independent Uniform(0, 1) random variables. What is the probability that the absolute difference |X − Y| is between 3/5 and 4/5?
A neighborhood has 20 adults and 11 children. Adults spend on average 80 minutes walking per week with standard deviation 10 minutes. Children spend on average 50 minutes per week with standard deviation 20 minutes. Assuming independence across people, what is the standard deviation of the total weekly minutes walked in the neighborhood?
A town has yellow and green buses. Yellow buses are late 1/6 of the time; green buses are late 1/8 of the time. Sophie boards a bus uniformly at random and is late. If the probability she rode a green bus given she was late is 1/3, what is the ratio of yellow to green buses in the town?
There are bags containing 2, 3, 4, 5, and 6 candies (one bag of each). Charlie independently takes each bag with probability 1/2. What is the probability Charlie takes at least half of all candies available?
Each day for one week (7 days), Taylor independently chooses eggs or yogurt for breakfast with equal probability. What is the probability there is a stretch of at least 5 consecutive days where Taylor only eats eggs?
A ~ Uniform[1, 2]. Conditional on A = a, B ~ Uniform[1, a^2]. What is E[B]?
Fatima has cards numbered 1 to 5, shuffled uniformly at random. She observes the first two cards differ by 1. What is the probability the last card is a 2?
You have an unfair coin with P(H) = 1/6; your opponent’s coin has P(H) = 1/3. You alternate flipping your respective coins until the first heads appears. You go first. What is the probability you get heads before your opponent?
Four friends independently and uniformly choose one of 4 restaurants. What is the probability that either all four end up at the same restaurant or that all four eat alone (i.e., all choose distinct restaurants)?
Two teams play a “first to three wins” series (best-of-5). Each game is fair (P(win)=1/2 for each team), independent across games. What is the expected difference in the final win counts between the tournament winner and loser?
A frog attempts two successive jumps to escape a hole: the first jump succeeds with probability 0.2; if that fails, a second jump succeeds with probability 0.3. If it fails the second jump, it falls and must try anew (starting again with the first jump). What is the expected total number of jumps until it escapes?

Solve probability and expectation problems

Quick Overview

Probability and Statistics Mini-Set

Solution

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Solve probability and expectation problems

Quick Overview

Probability and Statistics Mini-Set

Solution

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