Consider three independent rolls of a fair six-sided die. (a) What is the probability that the sum of the three rolls is 11? (b) What is the probability that the sum of the three rolls is 12? (c) In the seventeenth century, Galileo explained the experimental observation that a sum of 10 is more frequent than a sum of 9, even though both 10 and 9 can be obtained in six distinct ways. Can you retrace Galileo’s thinking?

a) 1/8 b) 1/12 c) probability of obtaining sum of 9 in a single roll of die without having the same number repeated twice is less than than of sum of 10  

Step-by-step explanation:

Probability of any digit in a single roll = 1/6

a) sum of 11 will be obtained if each roll has the following result

1 and 4 and 6, 1 and 5 and 5, 1 and 6 and 4, 4 and 1 and 6, 4 and 6 and 1, 6 and 4 and 1, 6 and 1 and 4, 5 and 1 and 5, 5 and 5 and 1, 2 and 5 and 4, 2 and 4 and 5, 4 and 2 and 5, 4 and 5 and 2, 5 and 2 and 4, 5 and 4 and 2, 2 and 6 and 3, 2 and 3 and 6, 3 and 2 and 6, 3 and 6 and 2, 6 and 3 and 2, 6 and 2 and 3, 3 and 3 and 5, 3 and 5 and 3, 5 and 3 and 3, 3 and 4 and 4, 4 and 3 and 3, 3 and 4 and 3

27(1/6 × 1/6 × 1/6)= 1/8

b) 2 and 4 and 6, 2 and 5 and 5, 4 and 2 and 6, 4 and 6 and 2, 6 and 4 and 2, 6 and 2 and 4, 5 and 2 and 5, 5 and 5 and 2, 2 and 6 and 4, 3 and 3 and 6, 3 and 6 and 3, 6 and 3 and 3, 3 and 4 and 5, 3 and 5 and 4, 4 and 3 and 5, 4 and 5 and 3, 5 and 3 and 4, 5 and 4 and 3

18(1/6 × 1/6 × 1/6)= 1/12

c) six ways of obtaining 10:  3+1+6, 3+2+5, 3+3+4, 4+2+4,4+5+1, 6+2+2

  six ways of obtaining 9: 1+2+6, 1+3+5, 1+4+4, 2+2+5, 2+3+4,3+3+3

To get 10, there are only two ways of repeating a number out of 6 ways. To get 9, there are three ways of repeating a number out of 6 ways

1) 2/5

2) zero=0/1

3) zero=0/2

4) -4/5

5) undefined=3/0