## Saturday, December 15, 2012

### Transition Matrices

This post is a part of a series of guest-posts on the applications of matrix multiplication. These posts were written by my pre-calc students:

Transition Matrices
by Leanna Krueger

Imagine that you own an apple orchard and you sell 3 different types of apples; Honeycrisp (A), Golden Delicious (B), and Gala (C). Honeycrisp apples sell the most while Golden Delicious is the runner up and Gala sells the least. You planted apple trees for each of these apples, and the cost of replacing each tree with the tree that produces apple A is too expensive until you get more money. In order to predict your earnings and decide when you can replant your orchard trees you need to estimate what people will buy. You know that this week the current market for apples is shown in the "state matrix" S shown below:
 A B C Percentage of People Buying 60% 30% 10%

Suppose you surveyed repeated buyers and found the following:
A.                If people are buying A this week, the probability of them buying A next week is 70%. The probability of them buying B next week is 20% and the probability for C is 10%.
B.                 If people are buying B this week, the probability of them buying A next week is 50%. The probability of them buying B next week is 30% and the probability for C is 20%.
C.                 If people are buying C this week, the probability of them buying A next week is 30%. The probability of them buying B next week is 40% and the probability for purchasing C again is 30%.
You could set this information up in a transition matrix, T, shown below:
 A B C A .7 .2 .1 B .5 .3 .2 C .3 .4 .3

If a family was buying apple B, and you wanted to know the probability of them buying apple A 4 weeks later you would have to multiply the 2 matrices above forming S*T. After multiplying them once, you would have to multiply them a second time, and third time, and a fourth time. In a sense, doing that is the same thing as finding S*T to the fourth power:
 A B C 1 week .6 .25 .15 2 weeks .59 .255 .155 3 weeks .587 .2565 .1565 4 weeks .5861 .25695 .156951

Therefore, the probability of the family buying apple A 4 weeks later is 58.61%.