Markov chain weather transitions

4 In a certain country, the weather each day is classified as fine or rainy. The probability that a fine day is followed by a fine day is 0.75 and the probability that a rainy day is followed by a fine day is 0.4 . The probability that it is fine on 1 April is 0.8 . The tree diagram below shows the possibilities for the weather on 1 April and 2 April.

1. Complete the tree diagram to show the probabilities. 1 April
   \includegraphics[max width=\textwidth, alt={}, center]{33c0bd01-f27b-424c-a78a-6f36178bc08c-08_601_405_706_408} 2 April Fine Rainy Fine Rainy
2. Find the probability that 2 April is fine.
   Let \(X\) be the event that 1 April is fine and \(Y\) be the event that 3 April is rainy.
3. Find the value of \(\mathrm { P } ( X \cap Y )\).
4. Find the probability that 1 April is fine given that 3 April is rainy.

7 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. 3
4. Write any calculations on page 5.
   \includegraphics[max width=\textwidth, alt={}, center]{091d6f43-ad01-4849-9f3c-3e58349aa169-4_2276_1490_324_363}

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]