# Worksheet: Stochastic Matrix

In this worksheet, we will practice using a matrix to represent probabilities.

Q1:

On any given day, an animal eats either cheese, grapes, or lettuce. The food the animal eats is influenced by the food it ate the previous day:

1. If it eats cheese today, tomorrow it will eat either lettuce or grapes, with equal probability, but not cheese.
2. If it eats grapes today, tomorrow it will eat grapes with a probability of , cheese with a probability of , or lettuce with a probability of .
3. If it eats lettuce today, tomorrow it will eat grapes with a probability of or cheese with a probability of , but not lettuce.

Represent these probabilities in the matrix , where the columns, in order, represent the events of eating cheese, grapes, or lettuce tomorrow; and the rows, in order, represent the events of eating cheese, grapes, or lettuce today.

• A
• B
• C
• D
• E

Letting , what is the interpretation of the entry ?

• Athe probability that the animal does not eat lettuce
• Bthe expected number of grapes the animal eats
• Cthe expected amount of cheese the animal eats
• Dthe probability that the animal eats grapes given that two days earlier the animal ate cheese
• Ethe probability that the animal eats cheese given that two days earlier the animal ate grapes

Q2:

What does it mean for a matrix to be stochastic?

• A has nonnegative entries and .
• B has nonnegative entries and .
• C
• D has negative entries and .

Q3:

Under which of the following conditions is the matrix a Markov matrix?

• A for all and
• B for all and
• C for all and
• D for all and