Probability of Rolling Exactly Three 6’s in Eight Die Rolls

When rolling a fair six-sided die, the probability of obtaining a specific number, such as a 6, is one possible area of interest. For instance, you might find yourself wondering: what is the probability of rolling exactly three 6's in eight rolls of a die? This article will delve into the mathematical concepts behind this question, offering a comprehensive explanation with step-by-step calculations.

Theoretical Approach

Let's begin by examining the theoretical underpinnings of this problem. Each roll of a die is an independent event, meaning the outcome of one roll does not influence the outcome of another. The sides of a standard die are numbered from 1 to 6, with a probability of 1/6 for rolling a 6 and 5/6 for not rolling a 6.

Calculating the Probability

When rolling a die multiple times, we can use the binomial distribution to determine the probability of a specific number of successes (in this case, rolling a 6) in a specified number of trials. In the context of dice, a "success" means rolling a 6, and a "failure" means not rolling a 6.

For the given problem:

N Number of trials 8 x Number of successes (rolling a 6) 3 p Probability of success (rolling a 6) 1/6 q Probability of failure (not rolling a 6) 5/6

The binomial probability formula is stated as:

P(C) N! / [x! (N-x)!] * p^x * q^(N-x)

Substituting the given values:

P(3 6’s) 8! / [3! (8-3)!] * (1/6)^3 * (5/6)^5

Step-by-Step Calculation

Let's break down the calculation step by step:

Calculate 8! / 3! 5! (1/6)^3 1/216 (5/6)^5 3125/7776 Multiply these values together:

8! / [3! (8-3)!] * (1/6)^3 * (5/6)^5 56 * (1/216) * (3125/7776)

Simplifying further:

56 * (1/216) * (3125/7776) 56 * 3125 / (216 * 7776) 175000 / 1679616 0.10416667

Verification through Combinatorial Approach

To verify the result, we can consider the combinatorial approach. The probability of any specific outcome in which exactly three of the eight rolls are 6’s can be computed by considering the following:

The number of ways to arrange three 6’s in eight rolls is given by the binomial coefficient: C(8, 3) 8! / [3! 5!] The probability of this specific arrangement is (1/6)^3 * (5/6)^5

Multiplying these values together confirms the result:

C(8, 3) * (1/6)^3 * (5/6)^5 56 * (1/216) * (3125/7776) 0.10416667

This result aligns with the previously calculated binomial probability.

Conclusion

In summary, the probability of rolling exactly three 6’s in eight rolls of a fair six-sided die is approximately 0.1041 or 10.42%. Understanding and applying these concepts can help in solving a wide range of probability problems involving dice and other discrete random events.

For a deeper dive into probability and statistics, or to verify the calculations, you can use software tools like PariGP, as shown in the initial snippet. These tools provide a practical method for checking the accuracy of your calculations and can be particularly useful in complex scenarios.