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.

1. 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?
2. 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?
3. 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?
4. 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?
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?
6. 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?
7. 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?
8. 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?
9. A ~ Uniform[1, 2]. Conditional on A = a, B ~ Uniform[1, a^2]. What is E[B]?
10. 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?
11. 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?
12. 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)?
13. 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?
14. 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?