Transition Matrices (DP IB Applications & Interpretation (AI)): Revision Note
Transition matrices
What is a transition matrix?
A transition matrix T shows the transition probabilities between the current state and the next state
The columns represent the possible current states
The rows represent the possible next states
The element of T in the ith row and jth column gives the transition probability tij of :
the next state being the state corresponding to row i
given that the current state is the state corresponding to column j
The probabilities in each column must add up to 1
The transition matrix depends on how you assign the states to the columns
Each transition matrix for a Markov chain will contain the same elements
The rows and columns may be in different orders though
E.g. Sunny (S) & Cloudy (C) could be in the order S then C or C then S
What is an initial state probability matrix?
An initial state probability matrix s0 is a column vector which contains the probabilities of each state being chosen as the initial state
If you know which state was chosen as the initial state then that entry will be 1 and the others will all be zero
You can find the state probability matrix s1 which contains the probabilities of each state being chosen after one interval of time
s1 = Ts0
s1 can be found from T and s0 using matrix multiplication
How do I find expected values after one interval of time?
Suppose the Markov chain represents a population moving between states
Examples include:
People in a town switching gyms each year
Children choosing a type of sandwich for their lunch each day
Suppose the total population is fixed and equals N
You can multiply the state probability matrix s1 by N to find the expected number of members of the population at each state after one interval of time
Similarly, multiplying the initial state probability matrix s0 by N will give the number of members of the population at each state initially
Examiner Tips and Tricks
If you are asked to find a transition matrix, check that all the probabilities within a column add up to 1. Drawing a transition state diagram can also help you to visualise the problem.
Worked Example
Each year Jamie donates to one of three charities: A, B or C. At the start of each year, the probabilities of Jamie continuing donate to the same charity or changing charities are represented by the following transition state diagram:
a) Write down a transition matrix 
for this system of probabilities.
b) There is a 10% chance that charity A is the first charity that Jamie chooses, a 10% chance for charity B and an 80% chance for charity C. Find the charity which has the highest probability of being picked as the second charity after the first year.
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