Probability Transition Matrices

Date: 02/10/99 at 04:50:53
From: Dee Skor
Subject: Probability matrix

Hi Dr. Math,

I have some questions on setting up transition matrices and using them 
to find probabilities. I would also appreciate it if you could explain 
how you are doing one or two of them.

Here are the questions:

1) There are 3 coffee brands that dominate the market: Brand A, Brand 
B, and Brand C. People switch from one brand to another all the time. 

If they use Brand A this week, there is 0.7 probability they will 
continue to use it next week, 0.2 probability they will switch to 
Brand B, and 0.1 probability they will switch to brand C.

If they are now using Brand B, there is 0.4 probability that they will 
switch to Brand A, 0.3 probability they will stay with Brand B, and 0.3 
probability they will switch to Brand C.

If they are now using Brand C, there is 0.2 probability they will 
switch to Brand A, 0.3 probability they will switch to Brand B, and 0.5 
probability they will stay with Brand C.

a) How can I express this as a transition matrix?

b) If a family is using Brand B, what is the probability they will be 
using Brand A 3 weeks later?

c) If a family starts with Brand B, what is the probability they will 
be using Brand C 4 weeks later?

2) Three people, John, Bill, and Peter, throw a ball to each other.
There is a probability of 1/3 that John will throw the ball to Bill.
There is a probability of 1/2 that Bill will throw the ball to Peter.
There is 1/4 probability that Peter will throw the ball to John.

a) How can I express this as a transition matrix?

b) Assuming the ball starts with Bill, what is the probability that he 
will have it back after 2 throws?

c) Assuming the ball starts with Peter, what is the probability Bill 
will have it after 3 throws?

4) If it rains today, there is a probability of 1/3 that it will rain 
tomorrow. If it doesn't rain today, there is a probability of 1/4 that 
it will rain tomorrow.

a) How do I set up a transition matrix?

b) If it doesn't rain on Wednesday, what is the probability that it 
will rain on Saturday?

Thanks in advance!
Dee

Date: 02/10/99 at 16:17:55
From: Doctor Anthony
Subject: Re: Probability matrix

In setting up the transition matrix, we set the columns to be "from" 
and the rows to be "to." This matrix represents the probabilities that 
we go from one situation to another situation in one step. To find the 
probabilities for n steps, we need to raise the original matrix to the 
nth power. This will be demonstrated in answering your questions.

Question 1:

The coffee transition matrix is

                 FROM
            A      B     C  
       A [0.7    0.4   0.2]
         |                |
   TO  B |0.2    0.3   0.3| 
         |                |
       C [0.1    0.3   0.5]

Note that the sum of each column is 1.
 
If a family is using Brand B, to find the probability they will be 
using Brand A 3 weeks later, we need to cube the above matrix:

                FROM
           A      B      C
      A [.541   .482   .436]
   TO B |.241   .254   .264|
      C [.218   .264   .300]   

So in 3 weeks the probability that someone will go from B to A is 
0.482. If a family starts with Brand B, to find the probability they 
will be using Brand C 4 weeks later, we raise the matrix to the 4th 
power. This gives us

                FROM
          A       B       C               
      A [.519   .492   .471]
   TO B |.246   .252   .256|
      C [.235   .256   .273]

Then the probability that someone starting with brand B will be using 
brand C 4 weeks later is 0.256.

Question 2:

Here is the ball-throwing transition matrix. Notice that to fill in the 
blanks, we use the fact that a person does not throw the ball to 
himself, and that the columns must sum to 1.

                     FROM
             John    Bill    Peter
        John[ 0      1/2      1/4]
   TO   Bill|1/3      0       3/4|
       Peter[2/3     1/2       0 ]

To find the probabilities after two throws, we need to square the 
matrix:

                       FROM
             John      Bill     Peter
       John [1/3       1/8        3/8]
   TO  Bill |1/2      13/24      1/12|
      Peter [1/6       1/3      13/24]

Note that in this case, none of the entries is 0 because there is a 
chance that the ball will come back to the original thrower in 2 
throws. So assuming the ball starts with Bill, the probability that he 
will have it back after 2 throws is 13/24.  

For three throws, we cube the matrix:

                       FROM
              John     BIll    Peter
        John [.292     .354    .177]  
   TO   Bill |.236     .292    .531| 
       Peter [.472     .354    .292]

Then the probability the ball will go from Peter to Bill in 3 throws is  
0.531.

Question 3:

The rain transition probability is

                   FROM
               Rain    No Rain
   TO    Rain [1/3        1/4] 
       No Rain[2/3        3/4]   

Each day is a "step". So if it doesn't rain on Wednesday, to find the 
probability that it will rain on Saturday (3 days later), we must cube 
the matrix:

                   FROM
               Rain    No Rain
   TO     Rain[.273     .273] 
       No Rain[.727     .727]

So going from no rain on Wednesday to rain on Saturday has a 
probability of 0.273. 

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/