IB DP Maths: AI HL復習筆記4.13.1 Markov Chains
Markov Chains
What is meant by a “state”?
- States refer to?mutually exclusive events?with the current event?able to change over time
- Examples of states include:
- Daily weather conditions
- The states could be: “sunny” and “not sunny”
- Countries visited by an inspector each day
- The states could be: “France”, “Spain” and “Germany”
- Store chosen for weekly grocery shop:
- The states could be: “Foods-U-Like”, “Smiley Shoppers” and “Better Buys”
- Daily weather conditions
What is a Markov chain?
- A?Markov chain?is a model that describes a?sequence of states?over a period of time
- Time is measured in discrete steps
- Such as days, months, years, etc
- Time is measured in discrete steps
- The?conditions?for a Markov chain are:
- The?probability?of a state being the?next state?in the sequence?only depends?on the?current state
- For example
The 11th?state?only depends?on the 10th?state
The first 9 states?do not affect?the 11th?state - This probability is called a?transition probability
- For example
- The?transition probabilities do not change?over time
- For example
The probability that the 11th?state is A given that the 10th?state is B is equal to the probability that the 12th?state is A given that the 11th?state is B
- For example
- The?probability?of a state being the?next state?in the sequence?only depends?on the?current state
- A Markov chain is said to be?regular?if it possible to reach any state after a finite period of time regardless of the initial state
What is a transition state diagram?
- A?transition diagram?is a?directed graph
- The?vertices?are the?states
- The?edges?represent the?transition probabilities?between the states
- The graph can contain
- Loops
- These will be the transition probabilities of the next state being the same as the current state
- Two edges between each pair?of vertices
- The edges will be in opposite directions
- Each edge will show the transition probability of the state changing in the given direction
- Loops
- The?probabilities?on the?edges coming out?of a vertex?add up to 1
Exam Tip
- Drawing a transition state diagram (even when the question does not ask for one) can help you visualise the problem
Worked Example
Fleur travels to work by car, bike or bus. Each day she chooses her mode of transport based on the transport she chose the previous day.
- If Fleur travels by car then there is a 40% chance that she will travel by car the following day and a 10% chance that she will travel by bike.
- If Fleur travels by bike then there is a 60% chance that she will travel by bike the following day and a 25% chance that she will travel by bus.
- If Fleur travels by bus then there is an 80% chance that she will travel by bike the following day and a 20% chance that she will travel by car.
Represent this information as a transition state diagram.
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