You’re a cowboy, and get involved in a three way pistol duel with two other cowboys. You are a poor shot, with an accuracy of only 33%. The other two cowboys shoot with accuracies of 50% and 100%, respectively. The rules of the duel are one shot per cowboy per round. The shooting order is from worst shooter to best shooter, so you get to shoot first, the 50% guy goes second, and the 100% guy goes third, then repeat. If a cowboy is shot he’s out for good, and his turn is skipped. Where or who should you shoot first?
The answer, of course, is to miss your first shot intentionally. I reasoned out the answer, but here’s the math, from a Berkeley discussion site.
You are A, 1/3 accuracy
B is 1/2 accurate
C is always accurate
x X y = x shoots at y
As A, you can either:
1) shoot at B
2) shoot at C
3) shoot at the ground (deliberately miss).
1) a X b
if you hit (1/3 of the time), you lose, and C fires next and takes you out
otherwise (2/3 of the time) b X c as C is B’s biggest threat. 1/2 the time B takes out C, which leaves a X b to resolve. 1/2 the time B misses, C kills B, and you have 1/3 chance to take out C before he takes you out.
First, a X b (with C out of the picture) is
1/3 + 2/3 * 1/2 * 1/3 + 2/3 * 1/2 * 2/3 * 1/2 * 1/3 + …
= 1/3 + (1/3)^n * 1/3
= 1/3 + 1/6 = 1/2
So a X b (with C there) is
2/3 * 1/2 * 1/2 + 2/3 * 1/2 * 1/3 = 10/36 chance of winning
2) a X c
if you hit C, B shoots are you, 1/2 the time you survive and it boils down to the a X b (C out of picture) calculated above.
if you miss C, B shoots at C, 1/2 the time hits and it boils down to a X b above. 1/2 the time B misses, C takes B out, and you have 1/3 chance to take C out.
So a X c is
1/3 * 1/2 * 1/2 + 2/3 * 1/2 * 1/2 + 2/3 * 1/2 * 1/3
= 1/12 + 1/6 + 1/9 = 13/36 chance of winning
This makes sense, relatively, as you best hope is to take out the most accurate shooter.
3) option 3, you skip your turn by firing into the ground.
b X c, 1/2 the time B hits and it becomes a X b calculated above. The other 1/2, C takes out B and you have 1/3 chance to take out C.
So, A miss on purpose is
1/2 * 1/2 + 1/2 * 1/3 = 1/4 + 1/6 = 15/36 chance of winning.
So your best option, at 15/36 = 41.7% of winning, is to miss your first shot on purpose!!