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Thread: Hit 'n miss

20030907, 13:32 #1
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Hit 'n miss
Alice, Bob and Carol agree to fight a pistol duel under the following unusual conditions. After drawing lots to determine who fires first, second and third, they take their places at the corners of an equilateral triangle. It is agreed that they will fire single shots in turn and continue in the same cyclic order until two of them are dead. At each turn, the person firing may aim wherever they please.
All three duelists know that Alice always hits the target, Bob is 80% accurate and Carol is 50% accurate.
Assuming that all three adopt the best strategy, and that no one is killed by a stray shot not intended for them, who has the best chance to survive?
A more difficult question is: What are the exact survival probabilities of the three duelists?
Alan

20030907, 17:32 #2
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Re: Hit 'n miss
I think the one most likely to live is TED since he is NOT participating!
Given that they will "adopt the best strategy" for survival, the Most likely survivor is <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>Carol (45%), followed by Bob (30%), and then Alice (25%), since Bob and Carol should try to kill the best shooter, Alice and Alice should go after Bob</font color=yellow></span hi>
The probabilities given are the overall, once you know the order of the shooting their is a different probablilty of success.
If Alice shoots first then <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>Bob has no chance, but Alice and Carol have equal chance</font color=yellow></span hi>
If Bob shoots first then <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>Carol is most likely to win followed by Bob, then Alice</font color=yellow></span hi>
If Carol shoots first then <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>Bob is most likely to win followed by Carol, then Alice</font color=yellow></span hi>
Alice's best chance of success <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>(50%) is when she shoots first</font color=yellow></span hi>
Bob's best chance of success <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>(63%) is when he is second to Carol</font color=yellow></span hi>
Carol's best chance of success <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>(55%) is when she is second to Bob</font color=yellow></span hi>
Alice's worst chance of success <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>(5%) is when she shoots LAST</font color=yellow></span hi>
Bob's worst chance of success <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>(0%) is when Alice shoots first</font color=yellow></span hi>
Carol's worst chance of success <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>(31%) is when she shoots first</font color=yellow></span hi>
Steve

20030908, 12:27 #3
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Re: Hit 'n miss
Hmmm <img src=/S/frown.gif border=0 alt=frown width=15 height=15>. I got quite different answers, because I interpretted "best strategy" differently.
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>I figured that C ought to let A & B fight it out first, by firing at nothing. C would then have first pop at the survivor of A & B. A & B would fire at each other, since each represented a bigger threat than C.</font color=yellow></span hi>
I'll have a closer look at your approach, then post mine.
BTW, Ted didn't qualify for the duel, since his name didn't start with D, and would have messed up the convenient A, B, C, ... sequence. <img src=/S/grin.gif border=0 alt=grin width=15 height=15>
Alan

20030908, 13:13 #4
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Re: Hit 'n miss
Concerning Ted, I just assumed this whole shootout had all come about from the aftermath of the wifeswapping!
What are you saying:
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow> Carol should do when it was her turn: NOT Shoot?.
The best approach is to always shoot the BEST shooter, since they have a better chance to kill you!
Carol's approach should be to SHOOT Alice since Carol has a BETTER chance (of surviving) against Bob. If she shoots and kills Bob, Alice will kill her! If she shoots and kills Alice, She has a 20% chance against Bob's shot.
I figure that a strategy to NOT shoot is never good. Neither Alice nor Bob will shoot at her in spite (ie since she shot at them) since that is NOT the best strategy for their survival. Alice MUST kill Bob first for the best chance and Bob MUST shoot at Alice to improve his chances.
As I said her best success is when Bob goes first and she goes 2nd. Bob will shoot at Alice. If Bob kills Alice she has a 50% chance of killing Bob. If Bob misses Alice (20%) she still has a 50% chance of killing Alice, and even if she misses Alice(50%), Alice will kill Bob, and she gets ANOTHER chance at Alice before Alice kills her.</font color=yellow></span hi>
Steve

20030908, 13:32 #5
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Re: Hit 'n miss
I was rethinking, your point and I believe that you are correct!
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>Carol SHOULD shoot to MISS if she goes first! That WILL increase her survival, since as I pointed out (BUT missed the subtlety), when she shot first was her WORST chance of success. Carol needs to turn the order if it is CAB to ABC (by missing) and if CBA to BAC by missing.
I will also have to look to see if her best strategy is ALWAYS to MISS if there are 2 around! This should NOT change my order: Carol will still have the best chance of success, though it will increase and the others will decrease. Alice's should decrease less than Bob's, so I don't know if that means that they will "flip" (I don't think so) </font color=yellow></span hi>
I will have to go over my numbers tonight (they are at home!) and I will post my modified calcs.
Steve

20030908, 14:18 #6
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Re: Hit 'n miss
I redid my calcs over lunch (much simpler with the new philosophy):
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow> breaks down in to only 2 cases since Carol always MISSES until there is ONLY one survivor:
When Alice shoots first Probabilities of survival:
Alice(50%), Bob (0%), Carol (50%)
When Bob Shoots first:
Alice(10%), Bob (35.6%), Carol (54.4%)
Overall that comes to:
Alice(30%), Bob (17.8%), Carol (52.2%)
So my ORIGINAL order is ALSO incorrect! It should be Carol, Alice, Bob!</font color=yellow></span hi>
Steve

20030908, 14:58 #7
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Re: Hit 'n miss
I agree with the strategy you propose, and also with the order of survival probabilities. I'm pretty sure my calcs (also not readily accessible at the moment) come up with the same figures  I did it in rational fractions so the percentages don't ring any immediate bells.
<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>I'll have to go over your method though, because I ended up summing an infinite geometric series to arrive at one of my answers. You seem to have avoided having to do this.</font color=yellow></span hi>
Alan

20030908, 15:15 #8
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Re: Hit 'n miss
I get the infinite geometric series also. But when you get something like 40 + 4 + 0.4 + 0.04 etc you can quit once you are going to report to 1 decimal place. You will get this whenever Alice dies first since she is the only DEFINITE kill.
Steve

20030909, 01:03 #9
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Re: Hit 'n miss
Checked my answers, and my fractions coincide with your percentages. So, what's the probability that we're both wrong? <img src=/S/grin.gif border=0 alt=grin width=15 height=15>
My answers are:<span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>
A 3/10
B 8/45
C 47/90
</font color=yellow></span hi>The infinite series, the one you mention, turns out to be an "easy" one, since it becomes the repeating decimal 0.4444... which is just 4/9.
The real catch here is that Ted the wife swapper was actually a mathematician in secret (what would his wifeswapping buddies have thought!) who wanted Carol on a permanent basis, so arranged the duel, having calculated the odds in advance.
Alan

20030909, 01:55 #10
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Re: Hit 'n miss
This could be used as a different puzzle. Given the shooting percentages you had in original puzzleand the original setup, instead of asking the survival rates, ask the question: If Carol shoots first, who should she aim at?
At this point we know the answer is <span style="backgroundcolor: #FFFF00; color: #000000; fontweight: bold"><font color=yellow>neither!</font color=yellow></span hi> But that is NOT obvious at first glance. I missed it and went thru a bunch of the figuring!
Steve