# Probability Transition matrix

4. Each day for lunch, Dr. Smith either brings in a lunch, has Taco Bell or gets a free lunch in the Springdale Room as he takes either students or guests of the department for lunch. On any given day there is a 10% probability that he will eat in the Springdale Room. If he brought in lunch yesterday, then he either remembers his lunch or goes to Springdale. If he had Taco Bell yesterday, then he has Taco Bell today half of the time. If he had Springdale yesterday, then he will always forger his lunch at home and be stuck with Taco Bell if there is not another Springdale lunch.

4a. Write out the probability transition matrix

4b. If Dr. Smith, just ate a lunch from home, then what is the probability that he will eat a Springdale lunch in 2 days?

4c. What proportion of time does Dr. Smith eat a home lunch?

4d. If Taco Bell costs $3 and a home lunch costs $1 and the Springdale room is free, then how much lunch budget should he allow for his semester, which includes 100 lunches?

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#### Solution Preview

a. Probability Transition matrix:

From

Spring Taco Home

Spring [.1 .1 .1

To Taco .9 .5 0

Home 0 .4 .9]

b. To find the probability in 2 days, we ...

#### Solution Summary

Probability Transition Matrix is evaluated clearly. The proportion of time Dr. Smith eats home lunches are provided.