If I go first, let me calculate my death probability.
This is basically the sum of an infinite geometric progression.
* I die by the 1st shot: 1/6
* Both survive after the first 2 shots and I die by the 3rd shot: (5/6)^2 * 1/6
* Both survive after the first 4 shots and I die by the 5th shot: (5/6)^4 * 1/6
...and so forth
The sum of the above: 1/6 * (1/(1-(5/6)^2))
= 1/6 * 1/(1-25/36) = 1/6 * 36/11 = 6/11
If I go first, let me calculate my death probability.
This is basically the sum of an infinite geometric progression.
* I die by the 1st shot: 1/6
* Both survive after the first 2 shots and I die by the 3rd shot: (5/6)^2 * 1/6
* Both survive after the first 4 shots and I die by the 5th shot: (5/6)^4 * 1/6
...and so forth
The sum of the above: 1/6 * (1/(1-(5/6)^2))
= 1/6 * 1/(1-25/36) = 1/6 * 36/11 = 6/11
If I go second, my death probability = 5/11