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:
Suppose you surveyed repeated buyers and found the following:
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%.
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