Probability Notes

1. Probability Notes Math 309 August 20

2. Some Definitions • Experiment - means of making an observation • Sample Space (S) - set of all outcomes of an experiment listed in a mutually exclusive and exhaustive manner • Event - subset of a sample space • Simple Event - an event which can only happen in one way; (or can be thought of as a sample point - a one element subset of S)

3. Since events are sets, we need to understand the basic set operations • Intersection - everything in A and B • Union - everything in A or B or both • Complement - everything not in A also denoted by a bar or a prime mark

4. You should be able to sketch Venn diagrams to describe the intersections, unions, & complements of sets. • Note that these set operations obey the commutative, associative, and distributive laws

5. DeMorgan’s Laws • Convince yourself that these are reasonable with Venn diagrams!

6. Another definition - A and B are mutually exclusive iff A  B = 

7. Axioms of Probability(these are FACT, no proof needed!) Let E represent an event, S the sample space, • Axiom 1: • Axiom 2: • Axiom 3: For pairwise mutually exclusive events, the probability of their union is the sum of their respective probabilities, i.e.

8. Sample Spaces with Equally Likely Outcomes • In an experiment where all simple events (sample points) are equally likely, one can find the probability of an event by counting two sets.

9. ComplementsUnionsIntersections

10. Theorems(You should be able to prove these using the axioms and definitions.) Let A and B be any two events. • . • Thm 7.1 • If , then

11. Unions get complicated if events are not mutually exclusive! P(A  B  C) = P(A) + P(B) + P(C) - P(A  B) - P(A  C) - P(B  C) + P(A  B  C) B

12. For mutually exclusive events the probability of their union is just the sum of their probabilities. However, recall

13. It is sometimes helpful to get mutuallyexclusive events by intersecting an event with another event and its complement. • For example, so that • Another helpful observation is that results in mutually exclusive events is:

14. Intersections & the multiplication rule Apply the multiplication rule to probabilities so that: • P(A  B) = P(A)*P(B|A) = P(B)*P(A|B) • P(B|A) is read, “the probability of B given A” • Tree diagrams may be helpful in visualizing this.

15. Intersections and intersection  multiply In general intersections get more complicated when there are more events, e.g. • P(ABCD) = P(A)* P(B|A)*P(C|AB)*P(D|A BC) We’ll see which type events become easy for intersections in a later section.

16. Combinatorial Methods Math 309 August 22

17. Combinatorics • Basic Principle of Counting • (a.k.a. Multiplication Principle) • Permutations • Permutations with indistinguishable objects • Combinations

18. Basic Counting Principle • If a choice consists of 2 steps where the first moutcomes and the second has n outcomes, then there are m*noutcomes for the whole choice. • The principle can be generalized for r steps. The number of outcomes of a choice with r stepsis the product of the number of outcomes of each step.

19. Permutations • # of arrangements of one set, order matters • application of the basic counting principle where we return to the same set for the next selection • P(n,r) = n!/(n-r)!

20. Combinations • the number of selections, order doesn’t matter • C(n,r) = n!/[(n-r)!r!] • the number of arrangements can be counted by selecting the objects and then ordering them • i.e. P(n,r) = C(n,r)*r!

21. Observations about Combinations • C(n, r) = C(n, n-r) • C(n, n) = C(n, 0) = 1 • C(n, 1) = n = C(n, n-1) • C(n, 2) = n(n-1)/2

22. Permutations with Indistinguishable Objects • Order the objects as if they were distinguishable • Then “divide out” those arrangements that look identical.

23. Combinations with Repetition • Select r objects from n objects when where each item can be selected more than once. • Add n-1 dividers to the r objects to be selected. In the r+n-1 “slots” select the location of the r items, C(r+n-1,r). The blank spaces will denote division of two types of objects.

24. Combining Counting Techniques • If we are careful with language, • when we say “AND”, we multiply • “AND”  multiplication  intersection • when we say “OR”, we add • “OR”  addition  union

25. Conditional Probability & Intersections Math 309 August 27

26. Conditional Probability P(A|B) • P(A|B) is read, “the probability of A given B” • B is known to occur.

27. Intersections and intersection  multiply In general intersections get more complicated when there are more events, e.g. • P(ABCD) = P(A)* P(B|A)*P(C|AB)*P(D|A BC)

28. Independent Events • A and B are independent if any of the following are true: • P(AB) = P(A)*P(B) • P(A|B) = P(A) • P(B|A) = P(B) • You need to check probabilities to determine if events are independent. • If A, B, C, & D are pairwise independent, P (AB C D) = P(A)*P(B)*P(C)*P(D)

29. Conditional Probability P(A|B) Formula • P(A|B) = P(A  B) / P(B), if P(B) > 0 • (Note that this is an algebraic manipulation of the formula for the probability of the intersection of 2 events.) • i.e. the conditional probability is the probability that both occur divided by what is given occurs