Compute probabilities and expectations in random processes

You are asked to solve the following probability/expectation questions. Unless stated otherwise, assume all random choices are uniform and independent.

1. Distinct digits : Choose an integer uniformly at random from 1 to 10,000 (inclusive). What is the probability that all digits in its decimal representation are distinct (no repeated digit)?
2. Stop when > 4 : Roll a fair six-sided die repeatedly until you roll a value greater than 4 (i.e., 5 or 6). What is the expected sum of all rolls?
3. 3-card rank product : From a standard 52-card deck, draw 3 cards uniformly without replacement. Map ranks as A=1, 2–10 as themselves, J=11, Q=12, K=13 . What is the expected value of the product of the three ranks?
4. Cancel H/T pairs : Toss 100 fair coins . Pairwise remove one Head and one Tail repeatedly (i.e., cancel H against T one-for-one). Let the number of coins left be the number of unmatched flips. What is the expected remaining number of coins ?
5. Stop on two identical in a row (sum) : Roll a fair die until you first see two consecutive rolls equal (e.g., 2 then 2). What is the expected sum of all rolls?
6. Run of 3 identical in 8 flips : Toss a fair coin 8 times (sequence order matters). What is the probability that the sequence contains at least one run of 3 consecutive identical outcomes (HHH or TTT anywhere)?
7. 1D random walk with die-based steps : Start at position 0. Each step, roll a die:
   • If the roll is 1,2,3: move right by that many units.
   • If the roll is 4,5,6: move left by (roll − 3) units (so steps are −1,−2,−3). What is the expected number of steps until your distance from the origin is at least 10 (i.e., |position| ≥ 10)?
8. Two 3-digit numbers difference is two-digit : Pick two integers independently and uniformly from 100 to 999 . What is the probability that their absolute difference is a two-digit number (i.e., between 10 and 99 inclusive)?
9. Two particles on an octagon : On a cycle graph with 8 vertices (an octagon), place two particles at opposite vertices (distance 4 along the cycle). Each second, independently for each particle, flip a fair coin to decide whether it moves one step clockwise or counterclockwise . What is the expected number of seconds until they occupy the same vertex?
10. Per-face running sums to 100 : Roll a fair die repeatedly. Maintain, for each face i∈{1,…,6}, a running sum S_i equal to the sum of all i’s rolled (equivalently S_i = i × count_i). Stop as soon as any S_i reaches at least 100 . At that stopping time, what is the expected number of even rolls (2,4,6) observed?
11. 13 cards with no aces : Draw 13 cards uniformly without replacement from a standard 52-card deck. What is the probability that the hand contains no aces ?
12. 2D walk to boundary of 10×10 grid : A particle starts at the center of a 10×10 grid (assume the natural discrete model where interior states form a 10×10 set and boundary states are the outer ring). Each second, flip two fair coins:

• HH: move North 1
• TT: move South 1
• HT: move West 1
• TH: move East 1 What is the expected time to hit the boundary?

13. Expectation of 2^(sum of 3 dice) : Roll 3 fair dice. Compute $\mathbb{E}[2^{X_1+X_2+X_3}]$ .
14. Stop on two identical in a row (product) : Roll a fair die until you first see two consecutive equal results. What is the expected product of all rolls?
15. Bankruptcy probability in 10 rolls : Start with $10. Roll a fair die exactly 10 times. If the roll is even, add the face value to your money; if odd, subtract the face value. What is the probability you go bankrupt at any time during the 10 rolls (money ≤ 0 at some step)?
16. No overlap in two length-3 samples (1..10) : You and a friend each independently draw 3 numbers from {1,…,10} with replacement . What is the probability that none of your numbers appears in your friend’s three draws?
17. Last die is 2 when exceeding 100 : Roll a fair die repeatedly until the running total sum first exceeds 100 . What is the probability that the last roll is exactly 2?