28 Jun '12 10:21

Rolling a standard six-sided die until a goal number > 1 is reached. How many ways are there do it? And is it so that there are as many ways involving an even number of rolls as an odd one, no matter what the target number is?

For example;

to reach 2; 1 way with an even # of rolls (1, 1) and 1 way with an odd # of rolls (2)

to reach 3; 2 ways each; (1, 2; 2, 1) ; (3; 1, 1, 1)

to reach 4; 4 ways each; (1, 3; 3, 1; 2, 2; 1, 1, 1, 1) ; (1, 1, 2; 1, 2, 1; 2, 1, 1; 4)

For example;

to reach 2; 1 way with an even # of rolls (1, 1) and 1 way with an odd # of rolls (2)

to reach 3; 2 ways each; (1, 2; 2, 1) ; (3; 1, 1, 1)

to reach 4; 4 ways each; (1, 3; 3, 1; 2, 2; 1, 1, 1, 1) ; (1, 1, 2; 1, 2, 1; 2, 1, 1; 4)