1. Transition matrix problem

Market research shows that if a person buys Brand X, the probability that they will buy Brand X next time is 0.75. If a person buys Brand Y, the probability that they will buy B the next time is 0.55.
Keith bought Brand X today.

a) Write the initial probability matrix and the transition matrix
b) what is the probability that keith will switch to Brand Y on his next purchase

My first reaction was for initial matrix Po= [ 0.75, 0.55] ???

for transition it would be
0.75 0.25
0.45 0.55

2. Re: Transition matrix problem

Originally Posted by terminator
Market research shows that if a person buys Brand X, the probability that they will buy Brand X next time is 0.75. If a person buys Brand Y, the probability that they will buy B the next time is 0.55.
Keith bought Brand X today.

a) Write the initial probability matrix and the transition matrix
b) what is the probability that keith will switch to Brand Y on his next purchase

My first reaction was for initial matrix Po= [ 0.75, 0.55] ???
The initial matrix is supposed to describe the initial situation. In this case, it should represent what Keith did today. What you are told about that is that Keith definitely bought Brand X, and definitely did not buy Brand Y. The numbers in the initial matrix should encode that information.

3. Re: Transition matrix problem

The initial matrix is supposed to describe the initial situation. In this case, it should represent what Keith did today. What you are told about that is that Keith definitely bought Brand X, and definitely did not buy Brand Y. The numbers in the initial matrix should encode that information.

So[ 0.50, 0.50] - initially?

4. Re: Transition matrix problem

No, the problem specifically says that "Keith bought Brand X today", not that he was equally likely to buy brand X or brand Y.

5. Re: Transition matrix problem

[1,0] initial one then??

0.75 0.25 - transition
0.45 0.55

If I multiply them I get 0.75 0.25

The answer in my book is 0.4?
Am I missing something?

6. Re: Transition matrix problem

You don't really need the matrix to answer the second part (it would be useful if they asked about the probability of buying brand X or brand Y on the fifth, or seventh or 100th time). You are told that "if a person buys Brand X, the probability that they will buy Brand X next time is 0.75".

Okay, Keith did buy brand x this time and so the probability he will buy it again the next time is 0.75. So either the text is wrong or you have quoted the exercise incorrectly.

I note that $A^7P_0$ has an X component of .403 which rounds to .4. Is it possible that the question not asking for the next time he purchases this kind of product but the 6th time (since he has already bought once)?

7. Re: Transition matrix problem

You don't really need the matrix to answer the second part (it would be useful if they asked about the probability of buying brand X or brand Y on the fifth, or seventh or 100th time). You are told that "if a person buys Brand X, the probability that they will buy Brand X next time is 0.75".

Okay, Keith did buy brand x this time and so the probability he will buy it again the next time is 0.75. So either the text is wrong or you have quoted the exercise incorrectly.

I note that $A^7P_0$ has an X component of .403 which rounds to .4. Is it possible that the question not asking for the next time he purchases this kind of product but the 6th time (since he has already bought once)?

8. Re: Transition matrix problem

You don't really need the matrix to answer the second part (it would be useful if they asked about the probability of buying brand X or brand Y on the fifth, or seventh or 100th time). You are told that "if a person buys Brand X, the probability that they will buy Brand X next time is 0.75".

Okay, Keith did buy brand x this time and so the probability he will buy it again the next time is 0.75. So either the text is wrong or you have quoted the exercise incorrectly.

I note that $A^7P_0$ has an X component of .403 which rounds to .4. Is it possible that the question not asking for the next time he purchases this kind of product but the 6th time (since he has already bought once)?