1. Warm-up Use the information below to find the population distribution after 20 years for the given population of giraffes. Age (in years)0-55-1010-1515-2020-25 Initial Population 1215582 Birthrate00.20.60.40 Survival Rate0.80.70.60.30

2. Warm-Up solutions

3. Markov Chains Matrices

4. Markov Chains Russian Mathematician A. A. Markov was recognized for his studies of linked chains of events Markov Chain – a process that arises naturally in problems that involve a finite number of events or states that change over time.

5. Example 1 If a student eats in the cafe on a given day, the probability that he/she will eat there again the next day is 70%. If a student does not eat in the cafe on a given day, the probability that he/she will eat in the cafe the next day is 40%. On Monday 75% ate in the cafeteria, and 25% did not. This is the initial probability of eating in the cafeteria. Let's make a tree! MONDAY Eat @ Cafeteria Eat elsewhere TUESDAY Eat @ Cafeteria Eat elsewhere Eat @ Cafeteria Eat elsewhere.25.75.70.30.40.60

6. Example 1 continued What percent of students will eat in the cafeteria on Tuesday? P(eat on Monday & eat on Tuesday) =.75 x.70 =.525 P(eat elsewhere Monday & eat on Tuesday) =.25 x.40 =.100 Percent of students eat in cafeteria on Tuesday.525 +.100 =.625 62.5%

7. Markov Chains Cafeteria Elsewhere Probability

8. Markov Chains Cafeteria Elsewhere FROM TO

9. There is a formula! In General: k = # of cycles (in this case days)

10. Example 1 Continued Cafeteria Elsewhere 62.5% eat on Tuesday Cafeteria Elsewhere 58.8% eat on Wednesday Cafeteria Elsewhere 57.3% eat on Fridayday

11. Example 2 A student has a class that meets Mon Wed and Fri. If he goes to class on one day, he goes to the next class with a probability of.5. If he doesn’t go to class that day, he goes to the next class with a probability of.75. Suppose the student attends the first class with a probability of.9. Draw A Tree:

12. Example 2 continued Initial Distribution Matrix D 0 : Transition Matrix T: What is the probability that he attends class on the third day? (k=2 days later)

13. Example 3 Central High School has uniforms. Each student must wear khakis and a polo that is either black, blue, or red. The discrete math class noticed several trends when it came to shirt selection, so they started to collect data. On the day they began their observation 30% of students wore black, 40% wore blue, and 30% wore red. Of the people who wore black initially, 10% wore black, 60% wore blue, and 30% wore red on the following day. Of the people who wore blue initially, 70% wore black, 2% wore blue, and 28% wore red on the following day. Of the people who wore red initially, 30% wore black, 40% wore blue, and 30% wore red on the following day. Draw a Tree.

14. Example 3 continued