# Chapter 18 - Think and Solve: 42

Let (a,b) represent rolling "a" with the first die and "b" with the second. Ways to roll a 2: (1,1) Ways to roll a 3: (1,2) and (2,1) Ways to roll a 4: (1,3) and (3,1) and (2,2) Ways to roll a 5: (1,4) and (4,1) and (2,3) and (3,2) Ways to roll a 6: (1,5) and (5,1) and (2,4) and (4,2) and (3,3) Ways to roll a 7: (1,6) and (6,1) and (2,5) and (5,2) and (3,4) and (4,3) Ways to roll a 8: (2,6) and (6,2) and (3,5) and (5,3) and (4,4) Ways to roll a 9: (3,6) and (6,3) and (4,5) and (5,4) Ways to roll a 10: (4,6) and (6,4) and (5,5) Ways to roll a 11: (5,6) and (6,5) Ways to roll a 12: (6,6)

#### Work Step by Step

Each of the 36 possibilities (called "microstates" in thermodynamics) is equally likely, but a total of seven is most likely to be observed because there are more combinations/microstates that give seven than any other total. Six of the 36 possible combinations result in a total of seven, showing that the probability of rolling a seven with a pair of fair dice is $\frac{1}{6}$.

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