Clay Pigeon Shooters
Clay pigeon shooting
is the art of shooting at special flying targets, known as clay pigeons or clay targets, with a shotgun. Suppose there are
clay shooters in a professional competition. Each shooter has equal ability and they all use identical shotguns. One throws
clay targets simultaneously, each shooter picks a target at random and shoots at it once, all the shooters shooting at the same time. The expected number of targets to get hit is
. Determine the value of
, the nearest integer of
.
The answer is 8.
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1 solution
Let X k = 1 if the kth target if hit and 0 otherwise. We want E ( ∑ i = 1 2 0 X k ) = ∑ i = 1 2 0 E ( X k ) = 2 0 E ( X 1 ) .
E ( X 1 ) = P ( T a r g e t 1 g e t s h i t ) = 1 − P ( N o s h o o t e r a i m s f o r t a r g e t 1 ) = 1 − ( 1 9 / 2 0 ) 1 0
So the answer if 2 0 ( 1 − ( 1 9 / 2 0 ) 1 0 ) = 8 . 0 2 5
You have to assume that all the shooters never miss.