11C Markov Chains and Transition Matrices

1. 11C Markov Chains and Transition Matrices A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. There are 200 members and on a particular day 150 use the gym and 50 do aerobics.

2. 11C Markov Chains and Transition Matrices A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. There are 200 members and on a particular day 150 use the gym and 50 do aerobics.

3. 11C Markov Chains and Transition Matrices A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. There are 200 members and on a particular day 150 use the gym and 50 do aerobics.

4. 11C Markov Chains and Transition Matrices A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. There are 200 members and on a particular day 150 use the gym and 50 do aerobics. day i+ 1 155 45 day i+ 2 155.544.5

5. 11C Markov Chains and Transition Matrices • A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. • There are 200 members and on a particular day 150 use the gym and 50 do aerobics. • What is the predicted long term usage of the gym and aerobics sessions? • let g represent gym users and a represent aerobics users

6. 11C Markov Chains and Transition Matrices Travellers can use the train or the bus to go to town. Research has shown that each month 20% of those using the bus switch to the train and 30% using the train switch to the bus. If, at the beginning of January, 4800 people were using the bus and 3600 were using the train, calculate: the number of people using the train at the beginning of May, the number of people using the bus and the train in the long term. January February - month i+ 1 3480 March - month i+ 2 4920 3420

7. 11C Markov Chains and Transition Matrices Travellers can use the train or the bus to go to town. Research has shown that each month 20% of those using the bus switch to the train and 30% using the train switch to the bus. If, at the beginning of January, 4800 people were using the bus and 3600 were using the train, calculate: the number of people using the train at the beginning of May, the number of people using the bus and the train in the long term. March April - month i + 3 3390 May - month i + 4 5025 3375

8. 11C Markov Chains and Transition Matrices • Travellers can use the train or the bus to go to town. Research has shown that each month 20% of those using the bus switch to the train and 30% using the train switch to the bus. • If, at the beginning of January, 4800 people were using the bus and 3600 were using the train, calculate: • the number of people using the train at the beginning of May, • the number of people using the bus and the train in the long term. • let brepresent bus users and trepresent train users • 5040

9. 11C Markov Chains and Transition Matrices A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. There are 200 members and on a particular day 150 use the gym and 50 do aerobics.

10. 11C Markov Chains and Transition Matrices A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. There are 200 members and on a particular day 150 use the gym and 50 do aerobics. Find: how many people will be attending the gym and aerobics classes on the 5th day, how many people will be attending the gym and aerobics classes in the long run. and

11. 11C Markov Chains and Transition Matrices A gym offers aerobic classes and gym sessions. Records show that 20% of members who use the gym will do aerobics the next day and 70% of members who do aerobics will use the gym the next day. There are 200 members and on a particular day 150 use the gym and 50 do aerobics. Find: how many people will be attending the gym and aerobics classes on the 5th day, how many people will be attending the gym and aerobics classes in the long run. and

12. 11C Markov Chains and Transition Matrices Netball Team. If the team wins, the probability of winning the next game is 0.75, if the team loses, the probability of winning the next game is 0.4. Find the probability, given they win the first game, that: they win 2 of their first 3 games, they win the 7th game, correct to 4 decimal places.

13. 11C Markov Chains and Transition Matrices Netball Team. If the team wins, the probability of winning the next game is 0.75, if the team loses, the probability of winning the next game is 0.4. Find the probability, given they win the first game, that: they win 2 of their first 3 games, they win the 7th game, correct to 4 decimal places.