Question

At a particular farmers market, they sell three kinds of fruit: strawberry, cherries, and blueberries. Assume that the farmers market has only one customer at a time. The owner estimates that strawberries are 50% likely to be followed by more strawberries, 15% likely to be followed by blueberries, and 35% likely to be followed by cherries. He also estimates that cherries are 45% likely to be followed by more cherries, 35% likely to be followed by strawberries, and 20% likely to be followed by blueberries. Lastly, he believes that blueberries are followed by more blueberries 45% of the time, and that strawberries are 25% likely to follow blueberries. For this problem, assume that the owner's estimated probabilities are exactly correct.

a) If the first fruit bought is strawberries, what is the probability that the 10th fruit bought will also be strawberries?

b) If the first fruit bought is strawberries, what is the probability that the first 10 fruits bought will all be strawberries?

Answer 1

Let the rows represent bying strawberries (state 0), cherries (state 1) and blueberries (state 2). Similarly the columns. Transition matrix is given by

(a) Given first fruit bought is strawberries (state 0) i.e then the tenth fruit is also strawberries is given by

.

we can solve it by two methods.

First method:

This is the first entry of the matrix . Using calculator we find

Hence , (ans)

second method :

If the stationary distribution after a long run is . Consider system of equations along with normality condition .

From this we thus have

For a unique solution we need three equations in three unknowns. Normality condition is mandatory. So, we can choose any  two equations from first three equations. Taking first two equations and the last equation and solving we get   . Hence probability of purchasing strawberries at tenth transition is approximately 0.3834.

(b) We need to find where   represent being in state j at time i. By using definition of markov chain we can write

If we assume probability of being in each state is equal i.e then