Answer to Hit or Get Hit

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Some basic assumptions on the rational strategy
1. All of them dont choose to deliberately miss everytime :) Else the game is not on and the probability of each surviving is 1.
2. Ugly and Bad will not shoot at each other and shoot at Good when he is alive. Because if they kill each other, then Good will kill them in the next shot
3. Good will shoot Bad when both Ugly and Bad are surviving. He wants to kill the one with higher shooting capability to increase his chances of survival

One wayy to attack this problem is conditional probability.
ugly shoots at Good and there are two conditions - Good survives or he gets killed.
Then under each scenario, there are more conditions and you go on and on and finish the problem. As you have already guessed that is the most difficult way.

Let us take an relatively easier route cutting down conditions as much as possible.
1. The first decision is to be made by Ugly - Shud he shoot or deliberately miss. We have already determined that if he shoots, he shud shoot at Good.
A) If he shoots Good and hits successfully, then both him and Bad remain in the contest with Bad getting the first shot.
Then the probability of Ugly's survival = p(Bad misses)*p(Ugly wins starting first) = 1/2*1/2 (see below for calc)
{ p(Ugly wins starting first) = p(Ugly hits bad) + p(Ugly misses)*p(Bad misses)*p(Ugly wins starting first) = 1/3 + 2/3*1/2*p(Ugly wins starting first)
Hence p(Ugly wins starting first) = 1/2 }
B) If he deliberately misses Good, then the chances of survival od Good, Bad and Ugly are as follows
p(Good survival) = p(Bad misses Good)*p(Good hits Bad)*p(Ugly misses Good) = 1/2*1*2/3 = 1/3
p(Bad survival) = p(Bad Hits Good)*p(Bad wins duel with Ugly, with Ugly starting first) = 1/2*1/2 = 1/4
p(Ugly survival) = 1-1/3-1/4 = 5/12

Since 5/12 is greater than 1/4, Ugly would choose to go with the strategy of deliberately missing the first shot. Once A misses the first shot, the probability of each survival are as given in option B
Chances of Good survival = 1/3
Chances of Bad survival = 1/4
Chances of Ugly survival = 5/12

The trick in this question is that the best has the least chance of survival. Note that the probabilities would have been the same if Bad had started. But the probabilities would change if Good had started.

Hope you enjoyed wracking the brains