高频题（0）
1. Comparative Analysis of Winning Probabilities
Given the same context as the previous question, without doing specific derivation, analyze which starting point is more likely to lead to a win: 3, 3, 3, 5, 7 or 3, 5, 5, 5, 7?
2. Optimal Strategy for Dice Game
With five uniform nine-sided dice, there are two ways to win: (1) three of a kind and two of a kind (aaabb), or (2) three consecutive numbers and two of the same number. The initial pattern is 3, 3, 5, 7, 8. You can choose one of the five dice to re-roll each time. What is the optimal strategy, and what is the expected number of rolls needed to win?
3. Probability of Median in a Range
Given three independent and identically distributed random variables that are uniformly distributed in the range [0,1], calculate the probability that the median of these variables falls within the interval [a, b], where 0 < a < b < 1.
4. Expected Return Calculation for Dice Game
You roll three fair 6-sided dice. If they all show the same number, you earn $20. If exactly two of the numbers are the same, you earn $10. If all of the numbers are different, you lose $2. Compute your expected return per roll in dollars.
5. Bakery Muffins Probability Calculation
Assume that in a bakery, each customer buys only one item at a time. There is a 70% chance a customer will buy a croissant and a 30% chance a customer will buy a muffin. There are only 2 muffins left and 5 people are still waiting in line. Compute the probability that these two muffins will be sufficient, i.e., no customer will want a muffin and find that there are none left.
