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The truel

Imagine it is 1872, you are sitting in a saloon with Smith and Brown, two of your old friends. However, the three of you do not seem to get along as you did before. As time passes, tall tales turn into boisterous arguments. At some point Smith has had enough and challenges both of you for a truel, a triangular pistol duel. You hesitate, knowing that you are the worst shot out of the three and look at Brown. You both know that Smith has the best aim and Brown comes second. In fact, from the many past duels, all three of you know your friends’, as well as your own exact odds of hitting a target: Smith never misses, Brown is 80 per cent accurate and you only hit half of your shots. Brown then agrees to the duel and so leaves you no choice but to agree as well, as refusing a duel brings deep shame. The three of you agree to take place in an equilateral triangle and fire single shots in turn until two of you are dead. At each turn, the man who is firing may aim at whoever he pleases. To determine who may fire first, second and third, you will draw lots. Now, assuming that all of you adopt the best strategy, who has the best chance to survive?  What are the exact survival probabilities of each of you?

It turns out that for a cowboy, you have great mathematical skills and you figure out the following unexpected result; you have the best chance of surviving, compared the other truelists, even though they have way better aim. The best strategy for Smith and Brown is to shoot at each other until one of them gets hit, as they form the greatest threat to the other. Therefore, at your turn, you plan to shoot up in the air if both of them are alive. As soon as one of them is shot, it will be your turn to shoot and you get a fair chance of winning. The exact individual survival probabilities are as follows:

For Smith, there is a \frac{1}{2} probability that he gets to make the first shot in his duel with Brown, in which case he kills him. Brown gets to shoot first at Smith also with probability \frac{1}{2}, but he then only has \frac{4}{5} probability of killing him. If Brown then misses, Smith will kill him. Smith’s probability of surviving this first duel with Brown is therefore \frac{1}{2}+\frac{1}{2}*\frac{1}{5}=\frac{3}{5} . If he manages to survive, you will get the next shot at him, which he survives with a probability of \frac{1}{2}. Smith’s overall odds of surviving are thus \frac{3}{10}

Browns odds turn out to be a little bit worse:

He has \frac{2}{5} probability of surviving Smith, after which he will face you. You will fire with 50% accuracy, and if you miss, he kills you with probability \frac{4}{5}. So his odds of killing you on his first shot after he killed Brown are \frac{1}{2}*\frac{4}{5} = \frac{4}{10}. Suppose you both miss again, however. He then gets another try and the probability of him killing you on his second turn in his duel with you, are \frac{1}{2}*\frac{1}{5}*\frac{1}{2}*\frac{4}{5}. = \frac{4}{100}. Summing up probabilities of all scenarios in which he kills you leads to 

\frac{4}{10} + \frac{4}{100} + \frac{4}{1000} = 0.4444= \frac{4}{9}. Combining this with the probability of Brown surviving Smith, leads to his overall probability of surviving: \frac{2}{5} * \frac{4}{9} = \frac{8}{45}.
For your probability of survival, simply subtract the other truelists’ probabilities from one to get

1-\frac{3}{10}-\frac{8}{45} = \frac{47}{90}. Even though you have the worst aim, you have the highest chance of surviving!


Dit artikel is geschreven door: Pieter Dilg

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