I have 27 hit points and my opponent has 50, and the winner is the player that reduces the other player's hit points to 0 first.
My expected damage inflicted per round is 5. My expected damage taken per round is 7/3.
I have the first attack.
What are my chances of winning?
edit: As pointed out, the question is unanswerable without further info on distributions, so here it is:
A round consists of multiple attacks. If I hit, I can attack again. My damage per hit is
5/6 1, and I have a 5/6 probability of hitting. I figured this makes a geometric series with $a = 5/6, r = 5/6$, so using $a / (1 - r)$ we get an expected damage of 5. Edit: Typo above. Damage is 1 point if I hit, so expected damage is 5/6. (Geometric series is still correct.)
When defending, similar rules apply. My damage taken is 7/6, and probability of being hit is 1/2. So a geometric series gives expected damage taken as 7/3. Edit to clarify: Actually I have a 1/6 chance of taking 3 damage, and 2/6 chance of taking 2 damage, which aggregates to 7/6.
So as a supplementary question, clearly this is a discrete distribution, but can we approximate to something linear and solve analytically? Or am I best off just doing a million trials by computer?