Probability and Expectation Practice (Take‑home)

Assume all coins and dice are fair unless stated otherwise. Show reasoning and final results.

1. Three coin flips: What is the probability all three flips are the same (all Heads or all Tails)?
2. One die thrown twice: What is the probability that the second throw differs from the first?
3. A 3‑flip "race" between Heads and Tails with score = (#Heads − #Tails): What is the probability that the side observed on the first flip is losing after all 3 flips?
4. A student walks from University (U) to Home (H), 4 blocks in a line. At each crossing they toss a coin: Heads → move forward one block; Tails → move backward one block (from U, Tails means staying put). Traversing a block takes 10 minutes; staying put takes 0. What is the probability they arrive in less than 45 minutes?
5. On average, how many die rolls are needed until all six faces have appeared at least once?
6. You roll a fair die repeatedly and track six running sums: the sum of all 1s, of all 2s, …, of all 6s. The game ends once any one of these six sums exceeds 100. What is the expected total sum of all dice rolled when the game ends?
7. Two people start at opposite corners of a 4×4 grid of nodes (coordinates 0..3 by 0..3). A moves only right or up uniformly at random among valid moves; B moves only left or down uniformly at random among valid moves. Both move every second. What is the probability they occupy the same node at some time?
8. From a shuffled 52‑card deck, how many cards do you expect to draw before the first Ace appears?
9. A 32‑team knockout tournament with teams ranked 1 (best) to 32 (worst); higher‑ranked always beats lower‑ranked. Seeds are assigned uniformly at random. What is the probability the final is Team 1 vs Team 3?
10. What is the probability that rolling six dice yields at least two distinct pairs (i.e., at least two different face values each occurring at least twice)?
11. A stock starts at $100 and changes by ±$ 1 daily based on a fair coin. What is the probability it equals $100 after exactly 10 days?
12. You repeatedly bet $1 on a fair coin, stopping upon reaching$ 20 or $0. Starting with$ 10, what is the probability you leave with $20?
13. You use a doubling strategy on a fair coin: bet $1, then$ 2, $4,$ 8, … until your first win or until your bankroll of $63 is exhausted. What is your expected profit?
14. A biased coin lands Heads with probability 2/3. You gain $1 on Heads and lose$ 1 on Tails. You start with $10; your opponent starts with$ 20 (total capital $30). Play continues until someone reaches$ 0. What is the probability you win?