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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June 9th, 2014 at 10:22 pm

Five. (Just a guess at this stage).

June 10th, 2014 at 2:06 am

I’d rather go with 6.513215599

June 10th, 2014 at 9:41 am

… so would I

June 10th, 2014 at 10:43 am

10 * (1 – 0.9^10)

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

June 11th, 2014 at 4:57 am

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)