1. CSE 3504: Probabilistic Analysis of Computer Systems Topics covered: Probability axioms Combinatorial problems (Sec. 1.5-1.8.3)

2. Probability axioms  Sample space:  Events:  Assign probabilities to events:  Example: A single coin toss

3. Probability axioms (contd..)

4. Probability axioms: Example  Sequence of three coin tosses  Compute the probability of event E1 – at least one head.  Compute the probability of event E2 – at most two heads.

5. Probability axioms: Example  System composed of CPU and memory  Sample space:  Events of interest – System up & system down:  Compute p(system up) and p(system down):

6. Formulating a probability model

7. Combinatorial problems  Ordered sample of size k with replacement

8. Combinatorial problems: Example  Ordered sample of size k with replacement (example)

9. Combinatorial problems: Example  Ordered sample of size k with replacement (example)

10. Combinatorial problems  Ordered sample of size k without replacement

11. Combinatorial problems (contd..)  Example: Ordered sample of size k without replacement

12. Combinatorial problems (contd..)  Example: Ordered sample of size k without replacement

13. Combinatorial problems (contd..)  Unordered sample of size k, without replacement

14. Combinatorial problems (contd..)  Example: Unordered sample of size k, without replacement

15. Combinatorial problems (contd..)  Example: Unordered sample of size k, without replacement