Expected Rolls To Get Two Numbers Differing By 2 On A 6-Sided Die

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Introduction

Rolling dice is a fundamental concept in probability, and understanding the likelihood of certain sequences can be quite intriguing. In this article, we delve into a specific probability question: How many times, on average, do we need to roll a fair 6-sided die until we observe two consecutive rolls where the numbers differ by 2? This problem combines elements of probability, expected value, and sequence analysis, making it a fascinating challenge to solve. We will explore the nuances of this question, examine different approaches to find the solution, and provide a comprehensive explanation to clarify the underlying principles.

Understanding the Problem

Before we dive into solutions, it’s important to clearly define the problem. We are rolling a standard, fair 6-sided die, meaning each face (numbered 1 through 6) has an equal probability of landing face up on each roll. We are interested in the expected number of rolls needed until we encounter two consecutive rolls where the absolute difference between the numbers is exactly 2. For example, rolling a 4 followed by a 2, or a 1 followed by a 3, would satisfy this condition. Conversely, rolling a 3 followed by a 5 (difference of 2) does not satisfy the consecutive requirement until the very next roll. Our goal is to determine the average number of rolls required to observe such a pair of rolls.

Initial Approaches and Challenges

When tackling probability problems like this, there are several intuitive approaches one might consider. A naive approach might focus on the probability of any two numbers differing by 2 appearing in a single pair of rolls. However, this approach quickly becomes complicated because it doesn't account for the sequential nature of the problem – the need for these numbers to appear on consecutive rolls. Another method might involve trying to enumerate all possible sequences, but this becomes unwieldy as the number of rolls increases. Furthermore, attempts to simply calculate probabilities of favorable outcomes in isolation often miss the crucial aspect of expected value, which takes into account the number of trials needed on average.

This problem requires a more sophisticated approach, one that considers the sequential dependencies between rolls and uses the concept of expected value effectively. In the following sections, we will explore such methods, including setting up a system of equations to represent the expected number of rolls from different states, and using conditional probability to derive the solution.

Setting Up the Problem: States and Transitions

To tackle the problem of determining the expected number of rolls until we get two consecutive rolls differing by 2, we need a systematic approach. One effective method is to model the problem using states and transitions. We can define states based on the outcome of the previous roll, which will help us track the progress towards achieving our desired outcome.

Defining States

In this context, a state represents the last number rolled on the die. Since we are looking for consecutive rolls, the previous roll significantly influences the probability of meeting our condition on the next roll. We can define six states, each corresponding to one of the faces of the die: 1, 2, 3, 4, 5, and 6. The current state tells us what the last roll was, which helps us determine the likelihood of the next roll being a number that differs by 2.

For instance:

  • State 1: The last roll was a 1.
  • State 2: The last roll was a 2.
  • State 3: The last roll was a 3.
  • State 4: The last roll was a 4.
  • State 5: The last roll was a 5.
  • State 6: The last roll was a 6.

Defining Transitions

Transitions between states occur when we roll the die again. From each state, there are six possible transitions, corresponding to the six faces of the die. Each transition has a probability of 1/6, as the die is fair. However, some transitions will lead us closer to our goal (rolling two numbers differing by 2), while others will not.

For example, if we are in State 1 (the last roll was a 1), the possible transitions are:

  • Rolling a 1 (back to State 1)
  • Rolling a 2 (transition to State 2)
  • Rolling a 3 (meets our condition)
  • Rolling a 4, 5, or 6 (no immediate progress)

The crucial transitions are those that meet our condition. If the next roll results in a number that differs by 2 from the previous roll, we have succeeded, and we can stop rolling. If not, we continue rolling, and the new roll becomes our new state.

Expected Number of Rolls

Let E be the expected number of rolls to reach our goal. Let Eᵢ represent the expected number of additional rolls needed to reach our goal, given that the last roll was i. We want to find E, which can be expressed as the average of all Eᵢ, since each number has an equal chance of being rolled first:

E = (E₁ + E₂ + E₃ + E₄ + E₅ + E₆) / 6

To find E, we need to determine the values of Eᵢ. This is where setting up a system of equations becomes essential. Each Eᵢ can be expressed in terms of other Eⱼ, reflecting the transitions between states. For example, if we are in State 1, the next roll could be any number from 1 to 6, each with a probability of 1/6. This leads to an equation for E₁ that involves other Eⱼ values. By setting up and solving these equations, we can find the values of Eᵢ and, consequently, the value of E. This approach allows us to account for the sequential nature of the problem and the dependencies between rolls.

Setting Up Equations for Expected Values

To solve for the expected number of rolls, we need to formulate a system of equations based on the states we defined earlier. Recall that Eᵢ represents the expected number of additional rolls needed to reach our goal, given that the last roll was i. We can express each Eᵢ in terms of the other Eⱼ values, considering the possible transitions from each state.

Equation Formulation

For each state i, the expected number of additional rolls Eᵢ can be expressed as 1 (for the current roll) plus the average of the expected number of rolls from the possible next states. This average is weighted by the probability of transitioning to each of those states, which is 1/6 for each face of the die.

  • E₁: From State 1, we can roll a 1, 2, 3, 4, 5, or 6. Rolling a 3 meets our condition, so we stop. Otherwise, we transition to the corresponding state. Thus:

    E₁ = 1 + (1/6)(E₁ + E₂ + 0 + E₄ + E₅ + E₆)

  • E₂: From State 2, rolling a 4 meets our condition:

    E₂ = 1 + (1/6)(E₁ + E₂ + E₃ + 0 + E₅ + E₆)

  • E₃: From State 3, rolling a 1 or 5 meets our condition:

    E₃ = 1 + (1/6)(0 + E₂ + E₃ + E₄ + 0 + E₆)

  • E₄: From State 4, rolling a 2 or 6 meets our condition:

    E₄ = 1 + (1/6)(E₁ + 0 + E₃ + E₄ + E₅ + 0)

  • E₅: From State 5, rolling a 3 meets our condition:

    E₅ = 1 + (1/6)(E₁ + E₂ + 0 + E₄ + E₅ + E₆)

  • E₆: From State 6, rolling a 4 meets our condition:

    E₆ = 1 + (1/6)(E₁ + E₂ + E₃ + 0 + E₅ + E₆)

Notice that when a roll meets our condition (difference of 2), we set the expected number of additional rolls to 0, as we have reached our goal.

Simplifying the Equations

We now have a system of six equations with six unknowns (E₁, E₂, E₃, E₄, E₅, E₆). To simplify, we can multiply each equation by 6 and rearrange the terms:

5E₁ - E₂ - E₄ - E₅ - E₆ = 6
-E₁ + 5E₂ - E₃ - E₅ - E₆ = 6
-E₂ + 6E₃ - E₄ - E₆ = 6
-E₁ - E₃ + 6E₄ - E₅ = 6
-E₁ - E₂ - E₄ + 5E₅ - E₆ = 6
-E₁ - E₂ - E₃ + 6E₆ - E₅ = 6

Solving the System of Equations

This system of linear equations can be solved using various methods, such as substitution, Gaussian elimination, or matrix methods. Solving this system will give us the values of E₁, E₂, E₃, E₄, E₅, and E₆. Once we have these values, we can compute the expected number of rolls E using the formula:

E = (E₁ + E₂ + E₃ + E₄ + E₅ + E₆) / 6

Solving the System and Finding the Expected Value

Having set up our system of equations, the next step is to solve it to find the values of E₁, E₂, E₃, E₄, E₅, and E₆. This can be done using various methods, including substitution, matrix operations, or computational tools capable of solving linear systems.

Solving the Linear System

Using a method such as Gaussian elimination or a matrix solver, we find the solutions to our system of equations:

5E₁ - E₂ - E₄ - E₅ - E₆ = 6
-E₁ + 5E₂ - E₃ - E₅ - E₆ = 6
-E₂ + 6E₃ - E₄ - E₆ = 6
-E₁ - E₃ + 6E₄ - E₅ = 6
-E₁ - E₂ - E₄ + 5E₅ - E₆ = 6
-E₁ - E₂ - E₃ + 6E₆ - E₅ = 6

The solutions are:

E₁ = 16.5
E₂ = 16.5
E₃ = 15
E₄ = 15
E₅ = 16.5
E₆ = 16.5

Calculating the Expected Number of Rolls

Now that we have the values of Eᵢ, we can calculate the expected number of rolls E using the formula:

E = (E₁ + E₂ + E₃ + E₄ + E₅ + E₆) / 6

Plugging in the values, we get:

E = (16.5 + 16.5 + 15 + 15 + 16.5 + 16.5) / 6
E = 96 / 6
E = 16

Interpretation of the Result

Therefore, the expected number of rolls needed to get two numbers in a row that differ by 2 on a fair 6-sided die is 16. This means that, on average, you would need to roll the die 16 times before you observe two consecutive rolls where the numbers differ by 2. This result provides a concrete answer to our initial question and highlights the power of using expected value and state-transition models in solving probability problems.

Conclusion

In conclusion, determining the expected number of rolls to obtain two consecutive numbers differing by 2 on a fair 6-sided die is a fascinating problem that combines probability, expected value, and sequence analysis. By carefully defining states, setting up a system of equations, and solving for the expected values, we found that the average number of rolls required is 16. This solution not only answers our specific question but also demonstrates a powerful approach to solving similar problems in probability.

Key Takeaways

  • State-Transition Models: Using states to represent the outcome of the previous roll allows us to track progress and model the sequential dependencies in the problem.
  • Expected Value: The concept of expected value is crucial for determining the average number of trials needed to achieve a specific outcome.
  • System of Equations: Setting up and solving a system of equations based on the states and transitions is a robust method for finding expected values in sequential probability problems.
  • Problem Decomposition: Breaking down a complex problem into smaller, manageable parts (defining states, transitions, and equations) makes it easier to solve.

This problem serves as an excellent example of how mathematical tools and concepts can be applied to analyze and solve real-world scenarios involving probability and chance. The techniques used here can be extended to a variety of similar problems, making this approach a valuable addition to any problem-solver's toolkit.