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Duck Shoot

Posted by Chris on June 8, 2014 – 6:19 am

Ten perfect shooters simultaneously and independently fire at ten ducks. On average, how many ducks are killed?

This post is under “Logic” and has 5 respond so far.

5 Responds so far- Add one»

1. 1. Wizard of Oz Said：

Five. (Just a guess at this stage).

2. 2. slavy Said：

I’d rather go with 6.513215599

3. 3. Chris Said：

… so would I

4. 4. Zorglub Said：

10 * (1 – 0.9^10)

The trick is to find the probability that a duck survives.

5. 5. Chris Said：

Two approaches. First: if only had 1 shooter, then the expected number of dead ducks would be 1. With 2 shooters, the expected number of dead ducks is 1 + 0.9, that’s because there’s a 90% chance that the first shooter will have aimed at a different duck to the second shooter. Similarly, for the third shooter there’s a 0.9*0.9 chance that neither of the first 2 shooters aims for the same duck as the 3rd shooter. Altogether, with 10 shooters, the expected (aka average) number of dead ducks is 1 + 0.9 + 0.9^2 +… + 0.9^9 = (1 – 0.9^10)/(1 – 0.9), see below.

Second approach: Consider a particular duck. The probability that none of the shooters will aim for it is 0.9^10. So the expected number of ducks not being shot at is 10 * 0.9^10. So the expected number being shot at is 10 – 10 * 0.9^10
= 10(1 – 0.9^10).

Either way we get 6.513… dead ducks.

s = a + ar + ar^2 + … + ar^n-1 => sr = ar + ar^2 + … + ar^n
=> s – sr = a – ar^n => s = a(1 – r^n)/(1-r)

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