6 The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).
- State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution.
- At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate \(\mathrm { P } ( 4 \leqslant X < 8 )\).
- At a busy time of the day, \(\lambda = 30\).
(a) Assuming that a Poisson distribution is valid, use a suitable approximation to find \(\mathrm { P } ( X > 35 )\). Justify your approximation.
(b) Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\).