Probability as a Shell Volume: Hitting a Ring on a Dart Board

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1. 問1

   Darts land with $c\,e^{-r^2}$ hits per unit area at distance $r$ from the centre. Using the method of shells, the expected number of hits in the disc $r < 0.6$ is $k\,c$. Find $k$ to three decimal places.

   $k =$

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2. 問2

   In the same dart model, $c\,e^{-r^2}$ hits per unit area, a total of $400$ darts land on the (idealised, unbounded) board. Find the constant $c$ to two decimal places.

   $c =$

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3. 問3

   In the dart model with $c\,e^{-r^2}$ hits per unit area, find the probability that a dart lands in the ring $0.3 < r < 0.9$, to three decimal places.

   $P =$

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4. 問4

   In the dart model with $c\,e^{-r^2}$ hits per unit area, find the radius $R$ of the disc $r < R$ that receives $75\%$ of the darts. Give $R$ to three decimal places.

   $R =$

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5. 問5

   A different thrower is modelled by $c\,e^{-r^2/2}$ hits per unit area. Using shells for the part and the whole, find the probability that a dart lands in the ring $1 < r < 3$, to three decimal places.

   $P =$

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