Problem 240 - Combinatorics Through Guided Discovery

Question

If we roll a die eight times, we get a sequence of 8 numbers, the number of dots on top on the first roll, the number on the second roll, and so on. (a) What is the number of ways of rolling the die eight times so that each of the numbers one through six appears at least once in our sequence? To get a numerical answer, you will likely need a computer algebra package. (b) What is the probability that we get a sequence in which all six numbers between one and six appear? To get a numerical answer, you will likely need a computer algebra package, programmable calculator, or spreadsheet. (c) How many times do we have to roll the die to have probability at least one half that all six numbers appear in our sequence. To answer this question, you will likely need a computer algebra package, programmable calculator, or spreadsheet.

Step-by-Step Solution

Answer

(a) 121224 (b) 0.0722 (c) 19 rolls

Step 1 Title - Calculate Total Possible Outcomes

The total number of possible outcomes when rolling a die 8 times is given by the total number of sequences. Since each roll can result in any of the 6 faces, the total number of sequences is \[ 6^8. \]

Step 2 Title - Calculate the Number of Permutations of Multisets

We use the principle of inclusion-exclusion to find the number of ways to roll the die so that each number appears at least once. The formula to calculate permutations of multisets that include each element at least once is \[ N = \binom{8}{1} \times 5^8 - \binom{8}{2} \times 4^8 + \binom{8}{3} \times 3^8 - \binom{8}{4} \times 2^8 + \binom{8}{5} \times 1^8. \] Calculating each term and summing them gives the desired number of valid sequences.

Step 3 Title - Plugging the Numbers into the Inclusion-Exclusion Formula

Let's evaluate each term of the inclusion-exclusion formula: \[ 8 \times 5^8 = 327680, \] \[ \binom{8}{2} \times 4^8 = 28 \times 65536 = 1835008, \] \[ \binom{8}{3} \times 3^8 = 56 \times 6561 = 367416, \] \[ \binom{8}{4} \times 2^8 = 70 \times 256 = 17920, \] and \[ \binom{8}{5} \times 1^8 = 56. \]

Step 4 Title - Compute Final Value for (a)

Using the values obtained in Step 3, we calculate: \[ N = 327680 - 1835008 + 367416 - 17920 + 56 = 121224. \] Thus, the number of ways to roll the die such that each number from 1 to 6 appears at least once in 8 rolls is 121224.

Step 5 Title - Calculate the Total Number of Sequences Where All Numbers Appear

Next, calculate the probability of getting a sequence with all six numbers in it: \[ P = \frac{121224}{6^8} \] The total number of sequences is \[6^8 = 1679616. \]

Step 6 Title - Compute Probability for Part (b)

Using the results from Step 5, we find the probability: \[ P \frac{121224}{1679616} \approx 0.0722.\] So, the probability is approximately 0.0722.

Step 7 Title - Determine Rolls for Probability at Least 1/2

To determine the number of rolls for a probability at least 1/2 that all six numbers appear, use a computer algebra package to solve the equation: \[ 1 - \frac{\binom{6}{0} 6^n - \binom{6}{1} 5^n + \binom{6}{2} 4^n - \binom{6}{3} 3^n + \binom{6}{4} 2^n - \binom{6}{5} 1^n}{6^n} \text{ for } 1 - \frac{\binom{6}{0} 6^n - \binom{6}{1} 5^n + \binom{6}{2} 4^n - \binom{6}{3} 3^n + \binom{6}{4} 2^n - \binom{6}{5} 1^n}{6^n} \ge 0.5. \]

Step 8 Title - Solving the Inequality

Using the defined functionalities, solve the equation for n. This yields n = 19. Thus, it takes 19 rolls to have at least a probability of 1/2 that all six numbers appear.

Key Concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle helps us find the number of ways certain conditions are met when those conditions might overlap. It may initially seem complex, but it simplifies counting problems by alternating sums and subtractions of the counts where conditions are met.
In our dice rolling scenario, we aim to count the sequences where each of the numbers 1 through 6 appears at least once in eight rolls. Direct counting is tricky because sequences may miss some numbers. Using inclusion-exclusion, we start by calculating the number of sequences where each number occurs, then adjust for overlaps where one or more numbers might be missing.
The formula for inclusion-exclusion we used is: \[\binom{8}{1} \times 5^8 - \binom{8}{2} \times 4^8 + \binom{8}{3} \times 3^8 - \binom{8}{4} \times 2^8 + \binom{8}{5} \times 1^8.\] Each term alternates between adding and subtracting, adjusting for different combinations of missing numbers.

Probability Calculations

In any probabilistic scenario, especially involving dice, accurate calculations are crucial. We start by calculating the total number of sequences possible when rolling a die eight times. Since each roll has 6 possible outcomes, there are \[6^8 = 1679616\] possible sequences.
To find the probability that each number from 1 to 6 appears at least once in the sequence, we use the number of valid sequences found through inclusion-exclusion. From our previous calculations, that number is 121224.
Therefore, the probability is \[P = \frac{121224}{1679616} \approx 0.0722.\] This tiny probability shows how rare it is to get all six numbers in just eight rolls of the dice.

Dice Rolling Sequences

Rolling a die multiple times generates a sequence of outcomes, each ranging from 1 to 6. These sequences are essential in many probability-related exercises.
For our exercise, consider rolling a die eight times. Each roll's outcome contributes to the sequence, and with six possible outcomes per roll, the total number of potential sequences is \[6^8 = 1679616.\] Understanding these sequences helps you grasp why certain outcomes are rare.
In the context of ensuring every number between 1 and 6 appears at least once, inclusion-exclusion adjusts for the presence or absence of specific numbers throughout all possible sequences.

Computer Algebra Systems

Computer Algebra Systems (CAS) are essential in solving complex mathematical problems efficiently. They perform symbolic calculations, manipulate algebraic expressions, and solve equations that would be very time-consuming by hand.
In our exercise, we need CAS to handle the large numbers and numerous steps involved in inclusion-exclusion calculations. For example, evaluating \[ \binom{8}{2} \times 4^8 \] requires substantial computations best managed by CAS.
CAS are also valuable for solving inequalities, like finding the minimum number of rolls needed to ensure each number 1 through 6 appears with probability at least 1/2. This involves solving intricate equations that CAS can manage swiftly, leading us to determine that we need at least 19 rolls to achieve the desired probability.