Can it be simpler than this?

Probability Level 4

Billy, Carly and David tried to find the location of a gold mine. As luck would have it, they all found it at the same time. But nobody wants to compromise and therefore decide to take turns shooting each other by Russian roulette. ( Just like old times ).All have fully loaded guns which won't run out of bullets and each is given a chance to shoot a bullet at a time. Billy starts the round, followed by Carly and finally David.They all know that Billy was the worst marksman who hit a target with half probability.Even though Carly was an experienced shooter who hits 7 out of every 8 targets, David was a professional who never missed a target. Suppose you were Billy. Use your brain and find the maximum probability that you will survive by using the best strategy.

Assumption: Everyone wants to survive in the best possible way!.
Give the first 3 decimal places of your answer. If the probability is 12.3%, enter your answer as 123.

The answer is 529.

1 solution

Alex Li
May 6, 2015

The optimal strategy is for Billy to miss his first shot intentionally. Note that Carly would attempt to eliminate David, because, if she eliminates Billy, she is guaranteed to lose. Now, since Billy will be fighting a duel in which he shoots first, he must have at least a $\frac{1}{2}$ probability of winning. However, if he eliminated one of his two opponents, his probability of winning the duel against the other person is less than $\frac{1}{8}$ . We calculate the probability of Billy winning if he misses:

Case 1: Carly hits David (probability $\frac{7}{8}$ ).

In this case, Billy and Carly will fight a duel, with Billy shooting first. The probability of Billy winning would be $\frac{1}{2}+\frac{1}{2}\times\frac{1}{16}+\frac{1}{2}\times(\frac{1}{16})^2+...=\frac{\frac{1}{2}}{1-\frac{1}{16}}=\frac{8}{15}$ , based on the formula for the sum of a geometric series.

Case 2: Carly misses (probability $\frac{1}{8}$ ).

In this case, David would shoot and eliminate Carly, because he would rather fight a duel with Billy that with Carly. Now, Billy and David fight a duel, with Billy shooting first. Billy must eliminate David on the first turn, or lose, so Billy has a $\frac{1}{2}$ probability of winning.

The overall probability of Billy winning is $\frac{7}{8}\times\frac{8}{15}+\frac{1}{8}\times\frac{1}{2}\approx0.529$ , so the final answer is $\boxed{529}$ .

I think it is worth stating explicitly that shooters cannot agree on who they are going to shoot, and that they can miss on purpose (or skip their turn, which is the same) which is an out of the box condition.

andrei vostrikov - 1 year, 5 months ago

Can it be simpler than this?

The answer is 529.

1 solution

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