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23 Jul 2003, 04:45
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Quote:
AkamaiBrah wrote:
Part 1:
Gunman X and gunman Y are in a duel. Let's say that gunman X will kill his target with one shot with probability P, and gunman Y will kill his target with one shot with probability Q. If gunman X shoots first, and they alternate shots until somebody dies, what is the probability that gunman X will survive?
Useful identity: if 0 < r < 1, then a*(1 + r + r^2 + r^3 + ... + r^infinity) converges to: a / (1 - r).
(A) P / (P + Q)
(B) P / ((1 - P)(1 - Q))
(C) P / (1 - (1 - P)(1 - Q))
(D) P / (1 - P * Q)
(E) P * (1 - Q)(1 - P)
C is the correct answer.
Part 1:
Gunman X and gunman Y are in a duel. Let's say that gunman X will kill his target with one shot with probability P, and gunman Y will kill his target with one shot with probability Q. If gunman X shoots first, and they alternate shots until somebody dies, what is the probability that gunman X will survive?
Useful identity: if 0 < r < 1, then a*(1 + r + r^2 + r^3 + ... + r^infinity) converges to: a / (1 - r).
(A) P / (P + Q)
(B) P / ((1 - P)(1 - Q))
(C) P / (1 - (1 - P)(1 - Q))
(D) P / (1 - P * Q)
(E) P * (1 - Q)(1 - P)
C is the correct answer.
Part 2.
Gladiators A, B, and C are engaged in deadly game. AFter drawing straws, the order of play is A, then B, then C, then A again and so on. A is given a gun with one bullet and he must shoot at a target. After A shoots, B is given a gun with one bullet and he must shoot at target, and so on. A is not a very good shot and has a 50-50 chance of killing his target when he shoots to kill. B is a sharpshooter and is guaranteed to kill his target. C is a good shot, and will kill his target 80% of the time. There are three guards, one is a sharpshooter who never misses, and the other two kill their target 80% of the time. In addition, the guards have automatic weapons, an unlimited number of bullets, and will fire immediately and rapidly if either shot upon, or if a gladiator tries to escape. Assuming that B and C will always choose their best strategy, what strategy should you recommend for A in order to maximize his chances of survival? (Please justify your answer -- the results of part 1 may come in handy.)
(A) Try to kill B
(B) Try to kill C
(C) Try to escape
(D) Try to kill the sharpshooting guard, then try to escape
(E) Shoot himself in the foot.
_________________
Best,
AkamaiBrah
Newport Beach
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23 Jul 2003, 05:27
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Hmmm...
Let's calculate chances for A to survive
A. kill B. --- A kills B, C misses, A kills C = 0.5*0.2*0.5=0.05
B. kill C. --- A kills C, B misses A = 0.5*0=0
C. escape --- The sniper guard kills A. Chances = 0
D. kill a sniper and escape. --- A kills sniper, the two other guard miss = 0.5*0.2*0.2=0.02
E. shoot his foot --- A shots his foot (does not affect the chances to survive), B kills C, A kills B = 1*0.5=0.5
E seems to be the best one.
Let's calculate chances for A to survive
A. kill B. --- A kills B, C misses, A kills C = 0.5*0.2*0.5=0.05
B. kill C. --- A kills C, B misses A = 0.5*0=0
C. escape --- The sniper guard kills A. Chances = 0
D. kill a sniper and escape. --- A kills sniper, the two other guard miss = 0.5*0.2*0.2=0.02
E. shoot his foot --- A shots his foot (does not affect the chances to survive), B kills C, A kills B = 1*0.5=0.5
E seems to be the best one.
23 Jul 2003, 05:33
Kudos
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I think the answer is A
Reason: Lets look at A's probability of survival under each choice
E is trivial
D) A succeeds and the other two guards miss = 0.5*0.2*0.2=0.02
C) 0 since sharpshooter guard will kill him
B) Even if A succeeds, B will kill him. A will survive only if he fails, B chooses to shoot at C and then A succeeds in killing B = 0.125
A) A survives if he succeeds in killing B, then C fails and then A succeeds in killing C OR if A fails, B chooses to kill C and then A succeeds in killing B = 0.175
Therefore, the probability of A's survival is maximum under choice A
Reason: Lets look at A's probability of survival under each choice
E is trivial
D) A succeeds and the other two guards miss = 0.5*0.2*0.2=0.02
C) 0 since sharpshooter guard will kill him
B) Even if A succeeds, B will kill him. A will survive only if he fails, B chooses to shoot at C and then A succeeds in killing B = 0.125
A) A survives if he succeeds in killing B, then C fails and then A succeeds in killing C OR if A fails, B chooses to kill C and then A succeeds in killing B = 0.175
Therefore, the probability of A's survival is maximum under choice A
nice try. think it through a little more carefully.... 
Okay, you've got me there ..... I can't think of an alternative approach... 
