The Expected Value of the Highest Number When Rolling n Fair 6-Sided Dice Simultaneously

In the realm of probability and statistics, one common question that arises is: if you roll n fair 6-sided dice simultaneously, what is the expected value of the highest number rolled? This article aims to explore and explain how to calculate this expected value, providing a comprehensive understanding of the underlying logic and methodology.

Introduction to the Problem

When dealing with a set of n fair 6-sided dice, the highest number rolled can range from 1 to 6. The expected value of this highest number, denoted as E[X], is a crucial measure in understanding the long-term average outcome of such dice rolls. To find E[X], we need to establish the probability distribution function for the highest number rolled, Pr(X k), and then apply the definition of expected value.

Step 1: Calculating the Probability

The first step in calculating the expected value is to determine the probability of the highest number being a given value k. We denote this probability as Pr(X k).

For k 1

The highest number is 1 only if all dice show 1. The probability is:

Pr(X 1) 1n/6n

For k 2

The highest number is 2 if at least one die shows 2 and no die shows a number larger than 2. The probability is:

Pr(X 2) (2n - 1n)/6n (1/3n - 1/6n)

For k 3

The highest number is 3 if at least one die shows 3 and no die shows a number larger than 3. The probability is:

Pr(X 3) (3n - 2n)/6n (1/2n - 1/3n)

For k 4

The highest number is 4 if at least one die shows 4 and no die shows a number larger than 4. The probability is:

Pr(X 4) (4n - 3n)/6n (2/3n - 1/2n)

For k 5

The highest number is 5 if at least one die shows 5 and no die shows a number larger than 5. The probability is:

Pr(X 5) (5n - 4n)/6n (5/6n - 2/3n)

For k 6

The highest number is 6 if at least one die shows 6. The probability is:

Pr(X 6) 1 - (5n/6n)

Step 2: Calculating the Expected Value

Using the probabilities calculated above, the expected value of the highest number can be found through the following formula:

E[X] 11n/6n) 21/3n - 1/6n) 31/2n - 1/3n) 42/3n - 1/2n) 55/6n - 2/3n) 6(1 - 5/6n)

Final Expression

Thus, the expected value E[X] can be calculated using the expression:

E[X] ∑ (k

This formula can be computed for any given n to find the expected value of the highest number rolled with n fair 6-sided dice. By applying this method, you can effectively determine the long-term average result of rolling multiple dice, providing insights into the distribution of the highest number rolled.