The number of minor accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(X\) having a Poisson distribution with mean 8.5.
Determine the probability that, in any particular year, there are:
at least 9 minor accidents;
more than 5 but fewer than 10 minor accidents.
The number of major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(Y\) having a Poisson distribution with mean 1.5.
Calculate the probability that, in any particular year, there are fewer than 2 major accidents.
The total number of minor and major accidents occurring each year at RapidNut engineering company may be modelled by the random variable \(T\) having the probability distribution
$$\mathrm { P } ( T = t ) = \left\{ \begin{array} { c l }
\frac { \mathrm { e } ^ { - \lambda } \lambda ^ { t } } { t ! } & t = 0,1,2,3 , \ldots
0 & \text { otherwise }
\end{array} \right.$$
Assuming that the number of minor accidents is independent of the number of major accidents:
state the value of \(\lambda\);
determine \(\mathrm { P } ( T > 16 )\);
calculate the probability that there will be a total of more than 16 accidents in each of at least two out of three years, giving your answer to four decimal places.