One Percent Risk: Calibrating the Dart Model from One Fact

Work each one out before you check it. Every question has a scratch pad and a calculator.

1. Question 1

   In the dart model, the probability that a dart lands in the ring $r_1 < r < r_2$ is $e^{-r_1^2} - e^{-r_2^2}$. For one thrower, a dart is $4$ times as likely to land inside the target $r < a$ as outside it. Find the value of $e^{-a^2}$.

   $e^{-a^2} =$

   Check

   Scratch padCalculator

2. Question 2

   In the same model ($P(r_1 < r < r_2) = e^{-r_1^2} - e^{-r_2^2}$), a thrower hits the target of radius $a$ half the time. Find the probability that a dart lands in the ring $a < r < 3a$, as a fraction.

   $P =$

   Check

   Scratch padCalculator

3. Question 3

   In the same model, a thrower has $e^{-a^2} = \tfrac13$ for a target of radius $a$. A spectator occupies the part of the ring $2a < r < 3a$ between 8 o'clock and 11 o'clock. Assuming the darts are spread evenly in angle, find the probability that one dart hits the spectator, as a percentage to two decimal places.

   $P =$%

   Check

   Scratch padCalculator

4. Question 4

   In the dart model with $c\,e^{-r^2}$ hits per unit area, find how many times more likely a dart is to land in the band $1 < r < 1.1$ than in the small disc $r < 0.1$ around the centre. Give the ratio to two decimal places.

   ratio $=$

   Check

   Scratch padCalculator

5. Question 5

   A thrower is modelled by $c\,e^{-r^2/18}$ hits per unit area. Find the radius $r > 0$ at which the hits per unit of $r$, the shell weight $w(r)$, is largest.

   $r =$

   Check

   Scratch padCalculator