NA3873 Lecture 4: Probability

1. NA387(3) Lecture 4: Probability (Devore, Ch. 2.2 � 2.2)

2. Topics Sample Spaces and Events Venn Diagrams � Unions / Intersections / Complements Probability Views Axioms and Interpretations of Probability Properties of Probability

3. Sample Spaces Sample Space Set of all possible outcomes of an experiment. Examples: Two Possible Outcomes: part is defective, not defective >2 Discrete Outcomes: rolling a die (1, 2, 3, 4, 5, 6) Continuous Variable Outcomes: length of a desk measurements (infinite number of possible outcomes) Sample spaces with multiple components Ex. 1: Textbook, two gas stations, six pumps each, how many in use? (0-6 possible for each station) Ex. 2: Textbook: Roll two dice, numbers on their tops. (get from Ex. 1?)

4. Other Examples: Sample Spaces COE Students, IOE Students Voters, TV Viewers Restaurants in Ann Arbor One day�s manufacturing output at one plant.

5. Events Subset of outcomes contained in the sample space. Simple Event � exactly one outcome Compound Event � more than one outcome If two possible outcomes ? binary event

6. Simple Event Unique outcome of an experiment Some examples of simple events: An engine is defective Flipping a coin, outcome: Tails Rolling a die, outcome: 3 Taking an exam, Getting an 85

7. Compound Event Suppose you have three generators to supply power, where each can be either working (W) or not working (N). Identify sample space of outcomes Identify a simple event Identify a compound event

8. Venn Diagrams In depicting multiple events, Venn diagrams are excellent visual tools.

9. Venn Diagram Example 1 A = Cars with SunroofsB = Cars with Air conditioning What does the shaded area represent ?

10. Venn Diagram Example 2 A = Cars with SunroofsB = Cars with Air conditioning What does the shaded area represent ?

11. Venn Diagram Example 3 A = Cars with SunroofsB = Cars with Air conditioning What does the shaded area represent ?

12. Tree Diagrams also useful Suppose you want to know how many 8 cylinder engines have automatic transmissions.

13. Set Theory Union of two or more Events Intersection of two (or more?) Events Complement of an Event Mutually Exclusive or Disjoint Events Collectively Exhaustive Events

14. Union of 2 Events A and B, denoted by the symbol A ??B, or A+B, is the event containing all elements that belong to A, B, or both.

15. Intersection of 2 Events A and B, denoted by the symbol,?A?? B, or just AB, is the event containing all elements that are common to A and B. Which area of this Venn diagram represents the intersection of these events?

16. Complement of an Event A is the subset of all elements of sample space (?)?that are not in A. Denoted as A' or Ac

17. Mutually Exclusive Events. Two events are mutually exclusive ( disjoint) if A ? B = �, that is, if A and B have no elements in common, I.e.; they cannot occur simultaneously, or the occurrence of the one prohibits the occurrence of the other. Draw a Venn Diagram that depicts two mutually exclusive events.

19. Objective of Probability Given an experiment and sample space, probability is used to assign a �degree of belief� that some event A will occur. Notation: P(A) or Pr(A) or Prob(A) = Probability that event A will occur Definition of Probability: Empirical, or Axiomatic Empirical: Ratio of favorable over total outcomes

21. ��Additive Rule� of Probability In the case that two events are not mutually exclusive, we can use the (intuitively obvious?) additive rule below:

23. Example: Throwing two Dice What is the probability of rolling a seven? What is the probability of rolling a seven or less? What is the probability of rolling an eleven or higher?

25. Interpreting Probability Objective (Empirical) Interpretation "If an experiment is conducted N times, and a particular attribute A occurs n times, then the limit of n/N as N becomes large is defined as the probability of the event A, denoted P(A).� -- (i.e., probabilities converge as n increases) E.g. when you say the prob of obtaining heads in a coin toss is 0.5, you are implying that over a large number of tosses, half the results will be heads. (assumes repeatable experiment) Subjective (Man-in-Street) Interpretation "The probability P(A) is a measure of the degree of belief one holds in a specified proposition A.� Example: we have a 70% chance of getting this contract. Subjective probabilities recognize that interpretations may vary based on circumstances and prior knowledge.

26. Probability Example 1 Prob Student has Visa Card = 0.5 Prob Student has MC = 0.15 Prob Student has Both Cards = 0.1. What is the probability that a student does not have a MC? What is the probability that a student has neither card? (Draw a Venn Diagram)

27. Probability Example 2 Given the following VENN diagram, What is the prob that events A and B will occur? What is the prob that events A and B and C will occur?