6 Anika walks along a street that contains parked cars. The number of cars that Anika passes, up to and including the first car that is white, is denoted by \(X\).
- State two assumptions needed for \(X\) to be well modelled by a geometric distribution.
Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Geo } ( p )\), where \(0 < p < 1\).
- For \(p = 0.1\), find \(\mathrm { P } ( X > 6 )\).
The number of cars that Anika passes, up to but not including the first car that is white, is denoted by \(Y\).
- For a general value of \(p\), determine a simplified expression for \(\mathrm { E } ( Y ) \div \operatorname { Var } ( Y )\), in terms of \(p\).
Ben walks along a different street that also contains parked cars. The number of cars that Ben passes, up to and including the first white car on which the last digit of the number plate is even is denoted by \(Z\).
It may be assumed that \(Z\) can be well modelled by the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } p \right)\), where \(p\) is the parameter of the distribution of \(X\).
It is given that \(\mathrm { P } ( \mathrm { Z } = 3 ) = \mathrm { kP } ( \mathrm { X } = 3 )\), where \(k\) is a positive constant.
- Determine the range of possible values of \(k\).