Three or more stages

6 Two bags contain coloured discs. At first, bag \(P\) contains 2 red discs and 2 green discs, and bag \(Q\) contains 3 red discs and 1 green disc. A disc is chosen at random from bag \(P\), its colour is noted and it is placed in bag \(Q\). A disc is then chosen at random from bag \(Q\), its colour is noted and it is placed in bag \(P\). A disc is then chosen at random from bag \(P\). The tree diagram shows the different combinations of three coloured discs chosen. \includegraphics[max width=\textwidth, alt={}, center]{5faf0d93-4037-4958-8665-1008477a79de-5_863_986_559_612}

1. Write down the values of \(a , b , c , d , e\) and \(f\). The total number of red discs chosen, out of 3, is denoted by \(R\). The table shows the probability distribution of \(R\).

   \(r\)	0	1	2	3
   \(\mathrm { P } ( R = r )\)	\(\frac { 1 } { 10 }\)	\(k\)	\(\frac { 9 } { 20 }\)	\(\frac { 1 } { 5 }\)

2. Show how to obtain the value \(\mathrm { P } ( R = 2 ) = \frac { 9 } { 20 }\).
3. Find the value of \(k\).
4. Calculate the mean and variance of \(R\).

3 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.

1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
2. Find the probability that
   (A) all three of these trains are on time,
   (B) just one of these three trains is on time,
   (C) the 1200 train is on time.
3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time.

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. [3 marks]
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.
   [8 marks]
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. [4 marks]