# 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:

- If it eats cheese today, tomorrow it will eat either lettuce or grapes, with equal probability, but not cheese.
- 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 .
- 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 eats cheese given that two days earlier the animal ate grapes
- Bthe expected amount of cheese the animal eats
- Cthe expected number of grapes the animal eats
- Dthe probability that the animal does not eat lettuce
- Ethe probability that the animal eats grapes given that two days earlier the animal ate cheese