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A fair 6-sided die is rolled four times in succession and the results are inserted into a list (a1, A2, A3, A4, A5, a6). What is the probability that a4 ≥ a5 ≥ a6?

A fair 6-sided die is rolled four times in succession and the results are inserted into a list (a1, A2, A3, A4, A5, a6). What is the probability that a4 ≥ a5 ≥ a6?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Probability with Dice Rolls - Understanding an Example** In this exercise, we explore the probability of a specific event occurring when rolling a fair, 6-sided die four times in succession. The outcomes of these rolls are recorded in a list as \((a_1, a_2, a_3, a_4, a_5, a_6)\). **Problem Statement:** A fair 6-sided die is rolled four times in succession and the results are inserted into a list \((a_1, a_2, a_3, a_4, a_5, a_6)\). What is the probability that \(a_4 \geq a_5 \geq a_6\)? **Explanation:** In this problem, we are interested in finding the likelihood that the value of the fourth roll is greater than or equal to the fifth roll, and the fifth roll is greater than or equal to the sixth roll. Given that each roll is independent, we can leverage our understanding of permutations and combinations to solve this. **Step-by-Step Approach:** 1. **Identify Outcomes:** Each of the rolls \(a_1, a_2, a_3, a_4, a_5, a_6\) can take any integer value from 1 to 6. 2. **Total Possible Combinations:** When rolling the die four times, there are a total of \(6 \times 6 \times 6 \times 6 = 6^4\) possible outcomes for the sequence. 3. **Desired Event:** The event of interest is \(a_4 \geq a_5 \geq a_6\) among the recorded values. Given this setup, the solution involves calculating the proportion of outcomes that satisfy the condition \(a_4 \geq a_5 \geq a_6\) out of all possible outcomes. This example enhances the understanding of probability concepts applied to dice, especially involving sequences and comparisons among rolled values. For a deeper dive into the specific calculations and enumerative combinatorics involved, further study into these fields of probability theory is recommended.
Transcribed Image Text:**Probability with Dice Rolls - Understanding an Example** In this exercise, we explore the probability of a specific event occurring when rolling a fair, 6-sided die four times in succession. The outcomes of these rolls are recorded in a list as \((a_1, a_2, a_3, a_4, a_5, a_6)\). **Problem Statement:** A fair 6-sided die is rolled four times in succession and the results are inserted into a list \((a_1, a_2, a_3, a_4, a_5, a_6)\). What is the probability that \(a_4 \geq a_5 \geq a_6\)? **Explanation:** In this problem, we are interested in finding the likelihood that the value of the fourth roll is greater than or equal to the fifth roll, and the fifth roll is greater than or equal to the sixth roll. Given that each roll is independent, we can leverage our understanding of permutations and combinations to solve this. **Step-by-Step Approach:** 1. **Identify Outcomes:** Each of the rolls \(a_1, a_2, a_3, a_4, a_5, a_6\) can take any integer value from 1 to 6. 2. **Total Possible Combinations:** When rolling the die four times, there are a total of \(6 \times 6 \times 6 \times 6 = 6^4\) possible outcomes for the sequence. 3. **Desired Event:** The event of interest is \(a_4 \geq a_5 \geq a_6\) among the recorded values. Given this setup, the solution involves calculating the proportion of outcomes that satisfy the condition \(a_4 \geq a_5 \geq a_6\) out of all possible outcomes. This example enhances the understanding of probability concepts applied to dice, especially involving sequences and comparisons among rolled values. For a deeper dive into the specific calculations and enumerative combinatorics involved, further study into these fields of probability theory is recommended.
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