1. CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Event algebra Probability axioms Combinatorial problems (Sec. 1.5-1.8.1)

2. Example  Sequence of three coin tosses:  Event E1 – at least two heads  Complement of event E1 – at most one head (zero or one head)  Event E2 – at most two heads

3. Example (contd..)  Event E3 – Intersection of events E1 and E2.  Event E4 – First coin toss is a head  Event E5 – Union of events E1 and E4  Mutually exclusive events

4. Example (contd..)  Collectively exhaustive events:  Defining each sample point to be an event

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

6. Probability axioms (contd..)

7. 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.

8. 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):

9. Formulating a probability model

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

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

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