2023 YIJC P2 Q9

A dart board is designed with an inner radius of \(r\) cm and an outer radius of \(15\) cm, as shown in the diagram below.

A dart throw is considered a ‘hit’ if it lands within the inner circle. It is assumed that any dart throw lands randomly on the dart board.

1. Show that the probability of a ‘hit’ is \(\frac{{{r}^{2}}}{225}\).[1]
2. Wei plays a round of \(8\) dart throws.
   1. The probability that Wei obtains more than six ‘hits’ is at most \(0.08\). Write down an inequality satisfied by \(r\). Hence find the range of possible values of \(r\).[2]
      It is now given that \(r=6\)
   2. Find the probability that Wei’s fifth throw is the second time that he obtains a ‘hit’.[2]
   3. Find the least number of throws that Wei needs to attempt such that the probability of obtaining at least \(1\) ‘hit’ from the attempts is at least \(0.7\).[3]
   4. Chan also plays a round of \(8\) dart throws. The number of ‘hits’ Chan obtained is more than the expected number of ‘hits’ Wei obtained. Find the probability that the number of ‘hits’ Chan obtained is fewer than twice the expected number of ‘hits’ Wei obtained.[3]