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The problem explores the expected number of rolls needed to see all sides of a die, extending the analysis from a single die to multiple dice rolled simultaneously. Understanding the nuances of probability and expectation is crucial in solving this problem, especially when considering the interplay between multiple dice. The initial intuition of simply dividing the expected rolls of a single die by the number of dice doesn't hold, revealing deeper statistical principles at play.

Understanding Expected Rolls

For a single six-sided die, the expected number of rolls to see each side at least once is calculated using the formula ##6(1/6 + 1/5 + 1/4 + 1/3 + 1/2 + 1) = 14.70##. This is a classic problem illustrating the concept of expected value in probability. Each term in the sum represents the expected number of rolls to see a new face, given how many distinct faces have already been observed. The formula for calculating expected rolls can be applied to many different scenarios.

When two dice are rolled simultaneously, one might intuitively expect the number of rounds to be half that of a single die. However, this intuition overlooks the constraint that rolls occur in pairs. The observed difference highlights the complexities introduced when considering multiple dice and their combined outcomes. The concept of expected rolls is fundamental in probability theory and has applications in various fields, including game theory and statistical analysis.

Markov Chains and Expected Rolls

The problem can be approached using Markov chains, where the state represents the number of distinct sides observed. Calculating the exit times from each state gives the expected number of rounds to see all six numbers. This method provides a more accurate approach than simply dividing the single-die expectation. Markov chains are powerful tools for modeling systems that evolve over time, with applications in physics, finance, and computer science.

The discrepancy between the intuitive calculation and the Markov chain result arises because the number of rolls is always an integer. As Dermot Craddock pointed out, if the last number appears on an odd-numbered roll, it's still necessary to complete the pair of rolls. This effect increases the expected number of rounds. The method of using Markov Chains helps solve the problem of expected rolls and can be used for more complex problems.

History-Dependent Analysis

A more detailed analysis requires considering the history of each round. Let ##S_k## be the total number of distinct numbers seen up to round ##k##, and ##X_k## be the number of distinct numbers rolled in round ##k##. The relationship ##S_k = X_1 + ... + X_k## does not always hold, as it only provides an upper bound. This is because some rolls might duplicate previously seen numbers, especially as more faces are observed.

When ##X_k = 1##, the probability that the two dice hit an already seen number is ##(1/6)((6 - S_{k-1})/6)##, in which case ##S_k## does not increase. Similarly, even when ##X_k = 2##, the sum might not increase if both dice show previously seen numbers. This history-dependent nature complicates the calculation but provides a more accurate estimate of the expected number of rounds. This analysis provides a more accurate view of expected rolls.

Initial Rounds Analysis

In the first round, ##R_1##, there's a ##5/6## probability of tossing two distinct numbers and a ##1/6## probability of tossing a double. Thus, ##S_1## is either 2 or 1 with these probabilities. Subsequent probabilities are conditional on these initial outcomes. Analyzing the initial rounds helps establish a foundation for calculating the overall expected value.

If ##S_1 = 1##, then in round ##R_2##, there's a ##(1/6)^2## probability that ##S_2 = S_1## (both dice show the same number). There's a ##2(1/6)(5/6)## probability that ##S_2 = S_1 + 1## (one die shows a new number), and a complementary probability that ##S_2 = S_1 + 2## (both dice show new numbers). This breakdown illustrates the complexity of tracking the number of distinct faces seen over multiple rounds. The concept of initial rounds analysis is important to understanding expected rolls.

Extrapolation and Constraints

Extrapolating the initial intuition to more dice leads to absurd results, such as expecting less than one roll to see all faces when using 15 dice. This highlights the flaw in the initial reasoning. Any additional constraint, such as requiring a prime number of rolls, increases the expected number of rolls. This is because constraints limit the ability to stop immediately after seeing all faces.

The fact that rolls always occur in pairs also contributes to the higher expectation. If the last number is seen on roll ##2k - 1##, the process continues to roll ##2k## times. This is equivalent to rolling the same number twice with a single die, which can be seen as wasted rolls. Understanding these constraints is crucial for a correct analysis of expected rolls.

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