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(a) What are the states of one standard (6-sided) die? What is the probability of each state occurring? (b) Consider a system of two standard dice. How many microstates are there? How many macrostates are there?

(a) What are the states of one standard (6-sided) die? What is the probability of each state occurring? (b) Consider a system of two standard dice. How many microstates are there? How many macrostates are there?

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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Please also form a general formula to calculate the total number of macrostates in terms of number of faces/sides and number of dice

**Exploring Probability with Dice** **(a) What are the states of one standard (6-sided) die? What is the probability of each state occurring?** A standard 6-sided die has six states, corresponding to the numbers on its faces: 1, 2, 3, 4, 5, and 6. The probability of each state occurring when the die is rolled is equal, thus each state has a probability of 1/6. **(b) Consider a system of two standard dice. How many microstates are there? How many macrostates are there?** When rolling two standard dice, there are a total of 6 × 6 = 36 possible outcomes or **microstates**. Each microstate represents a unique combination of the numbers on the two dice. The **macrostates** are defined by the sum of the numbers on the two dice, ranging from 2 (1+1) to 12 (6+6): - Sum 2: 1 microstate (1+1) - Sum 3: 2 microstates (1+2, 2+1) - Sum 4: 3 microstates (1+3, 2+2, 3+1) - Sum 5: 4 microstates (1+4, 2+3, 3+2, 4+1) - Sum 6: 5 microstates (1+5, 2+4, 3+3, 4+2, 5+1) - Sum 7: 6 microstates (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) - Sum 8: 5 microstates (2+6, 3+5, 4+4, 5+3, 6+2) - Sum 9: 4 microstates (3+6, 4+5, 5+4, 6+3) - Sum 10: 3 microstates (4+6, 5+5, 6+4) - Sum 11: 2 microstates (5+6, 6+5) - Sum 12: 1 microstate (6+6) **(c) What are the macrostates? How many microstates are there in
Transcribed Image Text:**Exploring Probability with Dice** **(a) What are the states of one standard (6-sided) die? What is the probability of each state occurring?** A standard 6-sided die has six states, corresponding to the numbers on its faces: 1, 2, 3, 4, 5, and 6. The probability of each state occurring when the die is rolled is equal, thus each state has a probability of 1/6. **(b) Consider a system of two standard dice. How many microstates are there? How many macrostates are there?** When rolling two standard dice, there are a total of 6 × 6 = 36 possible outcomes or **microstates**. Each microstate represents a unique combination of the numbers on the two dice. The **macrostates** are defined by the sum of the numbers on the two dice, ranging from 2 (1+1) to 12 (6+6): - Sum 2: 1 microstate (1+1) - Sum 3: 2 microstates (1+2, 2+1) - Sum 4: 3 microstates (1+3, 2+2, 3+1) - Sum 5: 4 microstates (1+4, 2+3, 3+2, 4+1) - Sum 6: 5 microstates (1+5, 2+4, 3+3, 4+2, 5+1) - Sum 7: 6 microstates (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) - Sum 8: 5 microstates (2+6, 3+5, 4+4, 5+3, 6+2) - Sum 9: 4 microstates (3+6, 4+5, 5+4, 6+3) - Sum 10: 3 microstates (4+6, 5+5, 6+4) - Sum 11: 2 microstates (5+6, 6+5) - Sum 12: 1 microstate (6+6) **(c) What are the macrostates? How many microstates are there in
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