Each summer the families of children in a school either stay at home (H) or go away on vacation (V). The activities for the holidays change according to the transition matrix below:
If 100 families stay at home in 2016, how many of these 100 families are expected to stay at home in 2017?
How many of these 100 families are expected to go on vacation in 2017?
How many of the families who stay home in 2017 are expected to stay at home in 2018 also?
A factory has 3 warehouses A, B and C. Forklift trucks are used to transport goods between the 3 warehouses. They start the day in one warehouse and end the day at the same or a different warehouse. The matrix, T, represents the transition matrix for this situation:
Calculate the percentage of forklift trucks which start the day at factory A and end the day at factory A.
There were 50 forklift trucks that started the day at factory B. Determine the number of forklift trucks expected to be parked at factory C at the end of the day. Round your answer to the nearest whole number.
A website uploads two blog posts each day, one about social issues \left ( S \right ) and the other about environmental concerns \left ( E \right ).
Of the people who read a blog every day, 52\% of those that read about social issues on one day will read about social issues the next day. Also, 83\% of those that read about environmental concerns will also read about environmental concerns the next day. Construct a state diagram that best represents this information.
Construct the transition matrix T that represents the transitional probabilities between each state.
On a certain day, the website records that 800 people read blog \left( E \right) while 550 people read \left( S \right). Use this information to predict the number of readers that will read blog \left( E \right) in 3 days time. Round your answer to the nearest whole number.
In a town with three supermarkets, the information below was gathered from a survey. The survey was conducted in October 2018. Assume that those customers that did not shop at a different store the following month, continued to shop at the same store.
42 \% of Store 1 customers will shop at Store 2 the following month.
45 \% of Store 1 customers will shop at Store 3 the following month.
39 \% of Store 2 customers will shop at Store 1 the following month.
45 \% of Store 2 customers will shop at Store 3 the following month.
39 \% of Store 3 customers will shop at Store 1 the following month.
6 \% of Store 3 customers will shop at Store 2 the following month.
Construct a transition matrix for the given information.
The market share at the time of the survey showed that 1200 customers shopped at Store 1, 800 customers shopped at Store 2 and 1000 customers shopped at Store 3. Construct the matrix which represents this information.
Find the number of customers expected to be shopping at each store in March 2019. Round your answers to the nearest whole number.
Calculate the long-term expected share of the customers shopping at each store as a percentage.
Sarah is planning an outdoor wedding at a particular location, and wants to know the likelihood of rain. Long-term data about the weather indicates that there is a 95\% chance that if it is dry one day, then the next day will also be dry. Conversely, there is a 44\% chance that a wet day will be followed by another wet day. This information is displayed in the table below:
Find the probability that it will rain in three days time if it is initially dry. Round your answer to three decimal places.
Find the long-term probability of rain if it is initially wet. Round your answer to three decimal places.
Sarah will only plan the wedding at this location if the long-term chance of rain is less than 10\%. Should she plan the wedding at this location?
| Today is dry | Today is wet | |
|---|---|---|
| Next day is dry | 0.95 | 0.56 |
| Next day is wet | 0.05 | 0.44 |
There are two supermarkets in a town, Millies and Reals. It is found that 50\% of shoppers who shop at Millies will shop there again the following week, and 65\% of shoppers who shop at Reals will shop there again the following week.
Using the first row and column for Millies, construct a transition matrix that represents this situation.
In one particular week, 400 people started shopping at Millies and 200 people started shopping at Reals. Using the first row for Millies, construct the initial state matrix for the situation.
By calculating the state S_1, determine how many people were shopping at Millies after 1 week.
By calculating the state S_2, determine how many people were shopping at Millies after 2 weeks.
By calculating the state S_5, determine how many people were shopping at Millies after 5 weeks. Round your answer to the nearest whole number, making sure to only round at the final step.
Determine the long-term prediction for the number of people that will shop at Millies.
At a particular university, students studying MATH101 will either attend a lecture in person or listen to a recording of the lecture later that week. It is found that 65\% of students who attend a lecture in person will attend the next lecture as well. It is also found that 85\% of students who listen to a recording instead will listen to a recording of the next lecture.
Using the first row and column for students who attend the lecture, construct the transition matrix T that represents the situation.
This year 300 students enrolled in MATH101, of which 270 attended the first lecture and the rest listened to a recording. Using the first row for students who attended the lecture, construct the initial state matrix for the situation.
Given that there are two lectures each week, and a semester goes for at least 13 weeks, approximate the steady state matrix for the system by calculating S_{25}. Round each element to the nearest whole number.
If only half of the 300 students attended the first lecture, and half listened to a recording, write down the new initial state matrix.
Approximate the steady state solution matrix for this new initial state matrix by calculating S_{25}. Round each element to the nearest whole number.
Students at an after school care program have two choices of activities to do in their free time, read a book (B) or play with an abacus (A). It was found that 25\% of children who read books one afternoon will play with the abacus the next afternoon, and 30\% of children who play with the abacus one afternoon will read books the next afternoon.
Construct the transition matrix for the situation outlined above.
If in one afternoon, 20 students play with the abacus and another 30 students read a book, how many will be expected to play with the abacus the next afternoon? Round your answer to the nearest whole number.
Explain why this model is unrealistic.
Irene (I) and Larry (L) are playing chess. They each have an equal chance of winning the first game. If Irene wins, then she gains confidence and her chance of winning the next game becomes 70\%. If Larry wins, his chance of winning the next game is 60\%.
Construct the initial probability matrix P_0 for the situation outlined above.
Construct the transition matrix T, which displays the probability of each player winning, given each player has already won.
Calculate P_1, the probabilty matrix after one game has been played.
Calculate the probability that Larry wins the second game.
Determine P_5, the probability matrix after five games have been played.
In the year 2019, there were 220\,000 people living in Town A and 60\,000 people in Town B. Each year, 5\% of the people in Town A move to Town B, and 23\% of the people in Town B move to Town A.
Construct the initial population matrix P_0.
Construct the transition matrix T that represents the shift in population.
Find the prediction of the population in each town in 2020.
Find the prediction of the population in each town in 2023.
Explain why this model is unrealistic.
Each year two stores, Supplies R Us (S) and The General (G) compete for business during the holiday period. They used surveys as a way of identifying how successful their marketing campaigns are. The surveys showed that 75\% of customers will return to Supplies R Us if they purchased goods there in the previous year, and 70\% of The General's customers will return to them year to year.
Construct a transition matrix T to represent the change in customers each year.
If both stores have 600 customers initially, find S_2, the matrix containing the expected number of customers that return to each store after two years.
Find S_7, the matrix containing the expected number of customers that return to each store after seven years.
A computer system can operate in two different modes. Every hour, it either remains in the same mode or switches to the other mode, according to the following initial transition probability matrix P:
Compute the two-step transition probability matrix, P^{2}.
If the system is in Mode 1 at 3:00 pm calculate the probability that it will be in Mode 1 at 8:00 pm. Round your answer to three decimal places.
Every year doctors come from around the world to attend a medical conference where they have the choice of two hotels to stay at, Andy’s (A) or Champion's (C).
The following is the initial transition probability matrix T:
\begin{matrix} & \begin{matrix} \\ \ A & \ C \end{matrix} \\ \begin{matrix}\ A \\ \ C \end{matrix} & \begin{bmatrix} 0.9 & 0.3 \\ 0.1 & 0.7 \\ \end{bmatrix} \end{matrix}A matrix recurrence relation used to determine the number of doctors expected to stay in each hotel (including gaining or losing doctors) is given below, along with the number of doctors that stayed at each hotel in 2017, S_0.
S_{n+1} = T \times S_{n} + \begin{bmatrix} 4\\ 5 \end{bmatrix} \text{ where } S_0=\begin{bmatrix} 70 \\ 30 \end{bmatrix}Calculate the number of doctors expected to stay at Andy’s in 2018.
Calculate the number of doctors expected to stay at Andy’s in 2020.
In a physical rehabilitation clinic, certain residents can choose out of 2 weekend activities each week, bowls (B) and golf (G). On the first weekend 70 people played bowls and 40 people played golf. It has been found that 70\% of residents who select bowls on one weekend will select bowls again the following weekend. It has also been found that 60\% of residents who select golf will select it again the following weekend.
Construct a transition matrix, T to represent this situation.
Construct the initial state matrix for the first weekend, S_1.
The participant numbers are also affected by residents who recover enough to move onto more physically demanding activities. 3 more people attend bowls each week and 5 more people attend golf each week. Construct matrix B to represent the number of people being added to the activities each week.
Construct the matrix recurrence relation which represents the expected number of residents selecting each activity on the second weekend, S_2 = TS_1+B.
Find S_2.
Complete the matrix recurrence relation which represents the expected number of residents selecting each activity on the third weekend, S_3 = TS_2+B.
Hence determine the expected number of people playing golf on the third weekend.
Every year doctors come from around the world to attend a medical conference where they have the choice of two hotels to stay at, Andy’s (A) or Champion's (C).
The following is the initial transition probability matrix T:
\begin{matrix} & \begin{matrix} \\ \ A & \ C \end{matrix} \\ \begin{matrix}\ A \\ \ C \end{matrix} & \begin{bmatrix} 0.9 & 0.3 \\ 0.1 & 0.7 \\ \end{bmatrix} \end{matrix}A matrix recurrence relation used to determine the number of doctors expected to stay in each hotel (including gaining or losing doctors) is given below, along with the number of doctors that stayed at each hotel in 2017, S_0.
S_{n+1} = T \times S_{n} + \begin{bmatrix} 4\\ 5 \end{bmatrix} \text{ where } S_0=\begin{bmatrix} 70 \\ 30 \end{bmatrix}Calculate the number of doctors expected to stay at Andy’s in 2018.
Calculate the number of doctors expected to stay at Andy’s in 2020.
In a physical rehabilitation clinic, certain residents can choose out of 2 weekend activities each week, bowls (B) and golf (G). On the first weekend 70 people played bowls and 40 people played golf. It has been found that 70\% of residents who select bowls on one weekend will select bowls again the following weekend. It has also been found that 60\% of residents who select golf will select it again the following weekend.
Construct a transition matrix, T to represent this situation.
Construct the initial state matrix for the first weekend, S_1.
The participant numbers are also affected by residents who recover enough to move onto more physically demanding activities. 3 more people attend bowls each week and 5 more people attend golf each week. Construct matrix B to represent the number of people being added to the activities each week.
Construct the matrix recurrence relation which represents the expected number of residents selecting each activity on the second weekend, S_2 = TS_1+B.
Find S_2.
Complete the matrix recurrence relation which represents the expected number of residents selecting each activity on the third weekend, S_3 = TS_2+B.
Hence determine the expected number of people playing golf on the third weekend.
Year 10 students are picking their electives for years 11 and 12. They have a choice between a language, L, and an arts subject, A. The probability that they choose either type of subject is affected by what type of elective they did in years 9 and 10. This is shown by the transition matrix, T, below:
\begin{matrix} & \begin{matrix} \\ \ L & \ A \end{matrix} \\ \begin{matrix}\ L \\ \ A \end{matrix} & \begin{bmatrix} 0.7 & 0.2 \\ 0.3 & 0.8 \\ \end{bmatrix} \end{matrix}Each year the number of students that choose each type of elective increases by 10. A matrix recurrence relation used to determine the number of students expected to choose each type of elective is given below, along with the number of students that chose each elective in 2020, S_0:
S_{n+1} = T \times S_{n} + \begin{bmatrix} 10\\ 10 \end{bmatrix} \text{ where } S_0=\begin{bmatrix} 120 \\ 80 \end{bmatrix}Find the number of students that choose a language in 2021.
Find the number of students that choose an arts subject in 2022.
A company is restructuring the accounting, A, engineering, E, and marketing, M, departments. The probability that the employees move within the same department or to another department is shown in the transition matrix below:
\begin{matrix} & \begin{matrix} \\ \ A & \ E & \ M \end{matrix} \\ \begin{matrix}\ A \\ \ E \\ \ M \end{matrix} & \begin{bmatrix} 0.6 & 0.2 & 0.3 \\ 0.3 & 0.8 & 0 \\ 0.1 & 0 & 0.7 \end{bmatrix} \end{matrix}Initially there were 16 people in accounting, 20 engineers, and 15 people in the marketing department. Write the initial state matrix S_0.
Each year, on average, 3 people leave the accounting department, 5 people leave the engineering department and 4 people leave the marketing department.
Write a recurrence relation to determine the number of employees in each department in the form S_{n+1}=TS_n-B.
Find S_2.
Hence, how many engineers will the company have in two years time?
A clothing store sells dresses, D, jackets, J, and pairs of pants, P. They are having a 2 for 1 sale where any two items can be bought for the price of one. The transition matrix below shows the probability of each type of item being chosen second, after choosing a particular first item.
\begin{matrix} & \begin{matrix} \\ \ D & \ J & \ P \end{matrix} \\ \begin{matrix}\ D \\ \ J \\ \ P \end{matrix} & \begin{bmatrix} 0.5 & 0.1 & 0.2 \\ 0.2 & 0.6 & 0.2 \\ 0.3 & 0.3 & 0.6 \end{bmatrix} \end{matrix}Due to the sale, each week they sell 8 more dresses, 10 more jackets, and 12 more pairs of pants than the week before. Before they started the sale they only sold 3 dresses, 8 jackets and 7 pairs of pants per week.
Find how many of each item they sold in their second week of the sale. Round your answers to the nearest whole number.
A group of mice are being studied. The age, initial population and average yearly birth rate and survival rate of the mice is shown in the table below:
| Age | Initial population | Birth rate | Survival rate |
|---|---|---|---|
| 1 | 12 | 0.6 | 0.5 |
| 2 | 8 | 2.2 | 0 |
Write the Leslie matrix, L, for this scenario.
Write the matrix for the initial population, S_0.
Write the recurrence formula that can be used to predict the number of mice in the form S_{n+1}=LS_n .
Find S_1.
Find S_2.
Find S_3.
Estimate the total number of mice that will be alive in 3 year's time.
A farmer has a number of chickens on his farm. The age, initial population and average weekly birth rate and survival rate of the chickens is shown in the table below:
| Age | Initial population | Birth rate | Survival rate |
|---|---|---|---|
| 1 | 3 | 5 | 0.9 |
| 2 | 4 | 4 | 0.6 |
| 3 | 2 | 3 | 0 |
Write the Leslie matrix, L, for this scenario.
Write the matrix for the initial population, S_0.
Write the recurrence formula that can be used to predict the number of chickens in the form S_{n+1}=LS_n .
Find S_1.
Estimate the total number of chickens the farmer will have in 1 week's time.
A ranch has a number of horses. The age, initial population and average yearly birth rate and survival rate of the horses is shown in the table below:
| Age group | Initial population | Birth rate | Survival rate |
|---|---|---|---|
| 0-10 | 4 | 0.6 | 0.8 |
| 11-20 | 6 | 0.7 | 0.5 |
| 21-30 | 1 | 0.4 | 0 |
Write the recurrence formula that can be used to predict the number of horses in the form S_{n+1}=LS_n , including initial population matrix S_0.
Find S_1.
Find S_2.
Estimate the total number of horses the farmer will have in 2 years time.
A reptile sanctuary has crocodiles living in it. The age, initial population and average yearly birth rate and survival rate of the crocodiles is shown in the table below:
| Age group | Initial population | Birth rate | Survival rate |
|---|---|---|---|
| 0-20 | 5 | 0.6 | 0.5 |
| 21-40 | 3 | 0.5 | 0.4 |
| 41-60 | 2 | 0.3 | 0 |
Write the recurrence formula that can be used to predict the number of crocodiles in the form S_{n+1}=LS_n , including initial population matrix S_0.
Find S_1.
Approximately many crocodiles were born in the first year?
A koala sanctuary has rescued 18 koalas that are currently living in it. The age, initial population and average yearly birth rate and survival rate of the koalas is shown in the table below:
| Age group | Initial population | Birth rate | Survival rate |
|---|---|---|---|
| 0-2 | 11 | 0.5 | 0.6 |
| 3-5 | 5 | 0.3 | 0.5 |
| 6-8 | 2 | 0.2 | 0 |
Write the recurrence formula that can be used to predict the number of koalas in the form S_{n+1}=LS_n , including initial population matrix S_0.
Assuming no other koalas are rescued, approximately many koalas will be living at the sanctuary in 3 years time?