Markov chain transition simulation

4 In a population colonizing an island 40\% of the first generation (parents) have brown eyes, \(40 \%\) have blue eyes and \(20 \%\) have green eyes. Offspring eye colour is determined according to the following rules. \section*{Eye colours of parents Eye colour of offspring} (1) both brown
(2) one brown and one blue \(50 \%\) brown and \(50 \%\) blue
(3) one brown and one green blue
(4) both blue \(25 \%\) brown, \(50 \%\) blue and \(25 \%\) green
(5) one blue and one green 50\% blue and \(50 \%\) green
(6) both green green

1. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of a parent randomly selected from the colonizing population.
2. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of offspring born of parents both of whom have blue eyes. The table in your answer book shows an incomplete simulation in which parent eye colours have been randomly selected, but in which offspring eye colours remain to be determined or simulated.
3. Complete the table using the given random numbers where needed. (You will need your own rules for cases \(( 2 )\) and 5 .)
   Each time you use a random number, explain how you decide which eye colour for the offspring. \(\square\)

6 In winter in Metland the weather each day can be classified as dry, wet or snowy. The table shows the probabilities for the next day's weather given the current day's weather. You are to use two-digit random numbers to simulate the winter weather in Metland.

1. Give an efficient rule for using two-digit random numbers to simulate tomorrow's weather if today is
   (A) dry,
   (B) wet,
   (C) snowy.
2. Today is a dry winter's day in Metland. Use the following two-digit random numbers to simulate the next 7 days' weather in Metland. $$\begin{array} { l l l l l l l l l l } 23 & 85 & 98 & 99 & 56 & 47 & 82 & 14 & 03 & 12 \end{array}$$
3. Use your simulation from part (ii) to estimate the proportion of dry days in a Metland winter.
4. Explain how you could use simulation to produce an improved estimate of the proportion of dry days in a Metland winter.
5. Give two criticisms of this model of weather.

4 The diagram represents a very simple maze with two vertices, A and B. At each vertex a rat either exits the maze or runs to the other vertex, each with probability 0.5 . The rat starts at vertex A .
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_79_930_534_571}

1. Describe how to use 1-digit random numbers to simulate this situation.
2. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex, and calculate the mean number of times it runs between vertices before exiting. The second diagram represents a maze with three vertices, A, B and C. At each vertex there are three possibilities, and the rat chooses one, each with probability \(1 / 3\). The rat starts at vertex A.
   \includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_566_889_1082_589}
3. Describe how to use 1-digit random numbers to simulate this situation.
4. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex.

2 Honor either has coffee or tea at breakfast. On one third of days she chooses coffee, otherwise she has tea. She can never remember what she had the day before.

1. Construct a simulation rule, using one-digit random numbers, to model Honor's choices of breakfast drink.
2. Using the one-digit random numbers in your answer book, simulate Honor's choice of breakfast drink for 10 days. Honor also has either coffee or tea at the end of her evening meal, but she does remember what she had for breakfast, and her choice depends on it. If she had coffee at breakfast then the probability of her having coffee again is 0.55 . If she had tea for breakfast, then the probability of her having tea again is 0.15 .
3. Construct a simulation rule, using two-digit random numbers, to model Honor's choice of evening drink given that she had coffee at breakfast. Construct a simulation rule, using two-digit random numbers, to model Honor's choice of evening drink given that she had tea at breakfast.
4. Using your breakfast simulation from part (ii), and the two-digit random numbers in your answer book, simulate Honor's choice of evening drink for 10 days.
5. Use your results from parts (ii) and (iv) to estimate the proportion of Honor's drinks, breakfast and evening meal combined, which are coffee. \section*{Question 3 begins on page 4}

6 Adrian and Kleo like to go out for meals, sometimes to a French restaurant, and sometimes to a Greek restaurant. If their last meal out was at the French restaurant, then the probability of their next meal out being at the Greek restaurant is 0.7 , whilst the probability of it being at the French restaurant is 0.3 . If their last meal out was at the Greek restaurant, then the probability of their next meal out being at the French restaurant is 0.6 , whilst the probability of it being at the Greek restaurant is 0.4 .

1. Construct two simulation rules, each using single-digit random numbers, to model their choices of where to eat.
2. Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at the French restaurant, and the proportion which are at the Greek restaurant. Adrian and Kleo find a Hungarian restaurant which they like. The probabilities of where they eat next are now given in the following table.

   \backslashbox{last meal out}{next meal out}	French	Greek	Hungarian
   French	\(\frac { 1 } { 5 }\)	\(\frac { 3 } { 5 }\)	\(\frac { 1 } { 5 }\)
   Greek	\(\frac { 1 } { 2 }\)	\(\frac { 3 } { 10 }\)	\(\frac { 1 } { 5 }\)
   Hungarian	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 3 }\)

3. Construct simulation rules, each using single-digit random numbers, to model this new situation.
4. Their last meal out was at the Greek restaurant. Use the random digits printed in your answer book to simulate their choices for the next 10 of their meals out. Hence estimate the proportion of their meals out which are at each restaurant. \section*{END OF QUESTION PAPER}