2 A plane flies regularly between airports D and T with an intermediate stop at airport M . The time of the plane's departure from, or arrival at, each airport is classified as either early, on time, or late. On \(90 \%\) of flights, the plane departs from D on time, and on \(10 \%\) of flights, it departs from D late. Of those flights that depart from D on time, \(65 \%\) then depart from M on time and \(35 \%\) depart from M late. Of those flights that depart from D late, \(15 \%\) then depart from M on time and \(85 \%\) depart from M late. Any flight that departs from M on time has probability 0.25 of arriving at T early, probability 0.60 of arriving at T on time and probability 0.15 of arriving at T late. Any flight that departs from M late has probability 0.10 of arriving at T early, probability 0.20 of arriving at T on time and probability 0.70 of arriving at T late.

1. Represent this information by a tree diagram on which labels and percentages or probabilities are shown.
2. Hence, or otherwise, calculate the probability that the plane:
   1. arrives at T on time;
   2. arrives at T on time, given that it departed from D on time;
   3. does not arrive at T late, given that it departed from D on time;
   4. does not arrive at T late, given that it departed from M on time.
3. Three independent flights of the plane depart from \(D\) on time. Calculate the probability that two flights arrive at T on time and that one flight arrives at T early.
   [0pt] [4 marks]