Three men, Bob, Dave and John have a quarrel about a woman. They decide to settl

Question

Three men, Bob, Dave and John have a quarrel about a woman. They decide to settle it with a three-way pistol dual, John hits his target 1/3 of the time, Dave hits the target 1/2 the time and Bob never misses. John gets the first shot. From then on whoever is shooting has to shoot at the most dangerous person (so Bob is the most dangerous, Dave is the next most dangerous and John is the least dangerous). What is John's best strategy? Prove what you just said by computing the probability that John will survive if on his first shot (i) He doesn't kill anybody (ii) He kills Dave (iii) He kills Bob

Explanation / Answer

(1)

John's best strategy is to doesn't kill anybody as it has the lowest probability of hitting the target and the best option is to leave so that Dave or Bob (who has greater probability of hitting the targets) has to choose among two targets on their chance and lowers the probability of John's losing his life.

(2)

John has 1/3 accuracy

Dave is 1/2 accurate
Bob is always accurate

Let the relaton X be defined as, x X y = x shoots at y  

i) He doesn't kill anybody.

So, John skip his turn by firing into the ground. So next option is
Dave X Bob, 1/2 the time Dave hits and it becomes John X Dave (see part (ii) for the probability calculation of John X Dave) . The other 1/2, Bob takes out Dave and John have 1/3 chance to take out Bob.

So, John miss on purpose is
1/2 * 1/2 + 1/2 * 1/3 = 1/4 + 1/6 = 15/36 chance of winning.

ii) He kills Dave

John X Dave
If John hit (1/3 of the time), Dave lose, and Bob fires next and takes John out
otherwise (2/3 of the time) Dave X Bob as Bob is Dave's biggest threat. 1/2 the time Dave takes out Bob, which leaves John X Dave to resolve. 1/2 the time Dave misses, Bob kills Dave, and John have 1/3 chance to take out Bob before he takes John out.

First, John X Dave (with Bob 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 (Using the sum of geometric series formula)

So John X Dave (with Bob there) is
2/3 * 1/2 * 1/2 + 2/3 * 1/2 * 1/3 = 10/36 chance of winning

iii) He kills Bob

John X Bob
If John hit Bob, Dave shoots John, 1/2 the time John survive and it comes down to the John X Dave (Bob out of picture) calculated above.
If John miss Bob, Dave shoots at Bob, 1/2 the time hits and it comes down to John X Dave above. 1/2 the time Dave misses, Bob takes Dave out, and John have 1/3 chance to take Bob out.

So John X Bob 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 John best hope is to take out the most accurate shooter.  
So John's best strategy is, at 15/36 = 41.7% of winning, is to miss his first shot on purpose!!

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