1 When a footballer takes a penalty kick the result is that either a goal is scored or a goal is not scored. It is known that, on average, a certain footballer scores a goal on \(85 \%\) of penalty kicks. During one practice session, the footballer decides to take penalty kicks until a goal is not scored. You may assume that the outcome of each penalty kick that the footballer takes is independent of the outcome of each other penalty kick. The random variable representing the number of penalty kicks up to and including the first penalty kick that does not result in a goal is denoted by \(X\).

1. State one further assumption that is necessary for \(X\) to be modelled by a Geometric distribution. For the remainder of this question you may assume that this assumption is valid.
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • Find the probability that the footballer takes exactly 3 penalty kicks.
   • Find the probability that the footballer takes at least 5 penalty kicks.