1. Probability I

2. Sample Spaces Sample space: The set of possible outcomes for a question or experiment Experiment: Flip a two sided coin Question: Did the North Korean government hack Sony?

3. Sample Spaces The number of outcomes in a sample space may be enormous Question: How many 5-card poker hands are possible? Question: How unique human DNA “profiles” are there?

4. Sample Spaces Outcomes can be continuous or discrete Discrete: Did he do it? Continuous: What is the mass of scheduled drugs seized?

5. Sample Spaces An event: Subset of a sample space E

6. Sample Spaces The complement to the event: Everything not in the event E E’E’

7. Sample Spaces A simple event is an event containing a single outcome. A compound event consists of more than one outcome. When the experiment is performed, if the outcome that occurs is in event E then we say E occurs.

8. Sample Spaces Example: Toss a coin: – Ω = {H,T}. – The event “head” is {H} and is a simple event. Example: Toss a coin twice – There are several possible sample spaces: – Ω = {0,1,2} : the number of heads appearing. – Ω = {HH,HT,TH,TT} : 1 st toss, 2 nd toss. The event {HT,TH} corresponds to getting exactly one head. It is a compound event.

9. Sample Spaces Example: Petraco et al. has built a database of common constituents in dust from varied and widespread locations: 539 categories of (semi)common components e.g.: "Human Hair Natural " " Head" "C Blk"

10. Venn diagram: A pictorial representation of combinations of sets making use of circles and rectangles. Some More Set Theory Language Empty set: The set containing no outcomes. The null set  or { }. Union: A  B occurs if A occurs, B occurs or both A and B occur. A B A  B

11. Some More Set Theory Language Intersection: A  B occurs if both A and B occur. Disjoint: A and B are disjoint or mutually exclusive if they have no outcomes in common, i.e. if A  B = . A B A  B A B

12. Kolmogorov Axioms of Probability Axiom 1: For any event A, Pr(A) ≥ 0 Axiom 2: Pr(Ω) = 1 Axiom 3: For a collection of mutually exclusive events, A 1, A 2, …, A n Everything else in probability theory can be deduced starting with these axioms

13. Kolmogorov Axioms of Probability Important consequences: A probability function assigns a probability to any event A such that: A partition of the sample space means: In words: The A i ’s chop up the sample space into non- overlapping (i.e. mutually exclusive) pieces.

14. Kolmogorov Axioms of Probability Important consequences: Probability of a complement Probability of nothing in the sample space

15. Kolmogorov Axioms of Probability Important consequences: Probability of a union of non-disjoint events In words: The probability of A or B is the probability of A plus the probability of B minus the probability of A and B Don’t count the probabilities of A and B twice if there is overlap between the events Three or more events are done similarly

16. Handy: DeMorgan’s Laws DeMorgan Law 1 DeMorgan Law 2 Kolmogorov Axioms of Probability

17. Example

18. Sally got shot by a purp. Let A = Joe shot Sally. Pr(A) = 0.3 Let B = Bill shot Sally. Pr(B) = 0.5 Draw a Venn diagram for this scenario Compute

19. Counting Formulas When various outcomes of an experiment are equally likely computing probabilities reduces to a counting problem Product rule for ordered k-tuples: k-tuple = (item 1, item 2, …, item k ) 2-tuple = a pair item 1 has n 1 possibilities, item 2 has n 2 possibilities, … Total number of ways to select k items = n 1 n 2 …n k

20. Example In a computer integers are represented as a pattern of 32 ones and zeros. How many possible “bit patterns” are there to represent integers in the scheme?

21. Example How many different 7-place license plates are possible if the first 3 are to be occupied by letters and the final 4 by numbers? How would this be different if repetition among letters and numbers is not allowed?

22. Counting Formulas How many ways are there to select r distinct items from a group of n distinct items? Permutations: If the order of selection is important Combinations: If the order of selection is irrelevant

23. Example Prof. Shenkin has 10 books that he wants to put on the shelf. Of these 4 are about probability, 3 are about algebra, 2 are about computers and 1 skydiving (Prof. Shenkin’s favorite hobby). He wants to arranges the books so that the same subjects are together on the shelf. They should be arranged by probability, algebra, computers and skydiving. How many different arrangements are possible? Is Prof. Shenkin OCD?

24. In R the following counts combinations and permutations: