Calculating the Probability of All Rolls Being the Same or All Different

In this article, we will explore the probability of rolling a standard six-sided die five times and obtaining either all the same or all different results. We will break the problem into two parts and combine the results to find the overall probability.

Introduction

A standard six-sided die is a cube with faces labeled 1 through 6. When rolling this die five times, we want to find the probability of obtaining either all the same or all different results.

Part 1: Probability of All Rolls Being the Same

To find the probability of all five rolls being the same, consider that each roll can result in any of the 6 faces. Here's the step-by-step reasoning:

Calculation

For the first roll, it doesn't matter which number is rolled because we need all rolls to be the same. For the second roll to be the same as the first, the probability is 1/6. For the third roll to be the same as the first two, the probability is also 1/6. Therefore, the combined probability for the first three rolls being the same is (1/6) x (1/6) 1/36. For the fourth roll to be the same as the first three, the probability is 1/6. So the combined probability for the first four rolls being the same is (1/6) x (1/6) x (1/6) 1/216. For the fifth roll to be the same as the first four, the probability is once again 1/6. Thus, the combined probability for all five rolls being the same is (1/6) x (1/6) x (1/6) x (1/6) 1/1296.

Thus, the probability of all five rolls being the same is:

Pall same 1/1296

Part 2: Probability of All Rolls Being Different

To find the probability of all five rolls being different, we need to choose 5 different numbers out of the 6 available and arrange them. This can be calculated as follows:

Calculation

The number of ways to choose 5 different numbers out of 6 is given by the binomial coefficient (binom{6}{5}), which is 6. The number of ways to arrange these 5 numbers is 5!, which is 120. The total number of favorable outcomes for all different rolls is 6 times 120 720. The total number of outcomes when rolling the die 5 times is 6^5 7776. Therefore, the probability of all five rolls being different is:

Pall different 720/7776 5/54

Part 3: Total Probability

The events of all rolls being the same and all rolls being different are mutually exclusive. Therefore, the total probability is the sum of these two probabilities:

Calculation

To add these fractions, we need a common denominator. The least common multiple of 1296 and 54 is 1296. We convert 5/54 to have a denominator of 1296: (frac{5}{54} frac{5 times 24}{54 times 24} frac{120}{1296}). Now we can add the fractions:

Pall same or all different (frac{1}{1296} frac{120}{1296} frac{121}{1296})

Final Result

Thus, the probability that the five rolls are either all the same or all different is:

(boxed{frac{121}{1296}})