Probability- Basic Concepts and Approaches

1. Probability and Statistics for Scientists and Engineers Probability-Basic Concepts and Approaches

2. Probability-Basic Concepts and Approaches • Basic Terminology & Notation • Basic Concepts • Approaches to Probability

3. Basic Terminology Definition – Experiment Any well-defined action. It is any action or process that generates observations. Definition - Outcome The result of performing an experiment

4. Basic Terminology Definition - Sample Space The set of all possible outcomes of a statistical experiment is called the sample space and is represented by S. Remark Each outcome in a sample space is called an element or a member of the sample space or simply a sample point.

5. Example An experiment consists of tossing a fair coin three times in sequence. How many outcomes are in the sample space? List all of the outcomes in the sample space.

6. Outcome Tree - Toss a fair coin three times 1 2 3 Outcomes H HHH HHT HTH HTT THH THT TTH TTT H T H H T T 0 H H T T H T T

7. Example An biased coin (likelihood of a head is 0.75) is tossed three times in sequence. How many outcomes are in the sample space? List all of the outcomes in the sample space.

8. Outcome Tree – Toss a biased coin three times 1 2 3 Outcomes H HHH HHT HTH HTT THH THT TTH TTT H T H H T T 0 H H T T H T T

9. Events Definition – Event An event is the set of outcomes of the sample space each having a given characteristic or attribute Remark An event, A, is a subset of a sample space, S, i.e., A S.

10. Example - Event Experiment Toss a fair coin 3 times in sequence Event Define the event A to be “2 heads occur” - How many outcomes of S are in A?

11. Events Continued Definition - Types of Events If an event is a set containing only one element or outcome of the sample space, then it is called a simple event. A compound event is one that can be expressed as the union of simple events. Definition - Null Event The null event or empty space is a subset of the sample space that contains no elements. We denote the event by the symbol .

12. Operations With Events Certain operations with events will result in the formation of new events. These new events will be subsets of the same sample space as the given events. Definition - The intersection of two events A and B, denoted by the symbol A B, or by AB is the event containing all elements that are common to A and B. S A A∩B B

13. Operations With Events Definition - Two events A and B are mutually exclusive if A  B = . Definition - The union of two events A and B, denoted by the symbol A  B, is the event containing all the elements that belong to A or to B or to both. S A A  B B

14. Operations With Events Definition - The complement of an event A with respect to S is the set of all elements of S that are not in A. We denote the complement of A by the symbol A´. Results that follow from the above definitions: A   = 0. A   = A. A A´ =  A A´ = S. S´ = . ´ = S. (A´) ´ = A. Venn Diagram S A A´

15. Probability Definition For any event A in S, the probability of A occurring is a number between 0 and 1, inclusive, i.e., where and where  is the null event.

16. Probability - Basic Questions • First, there is a question of what we mean when we say that a probability is 0.82, or 0.25. • What is probability? • Then, there is the question of how to obtain numerical values of probabilities, i.e., how do we determine that a certain probability is 0.82, or 0.25. • How is probability determined? • Finally, there is the question of how probabilities can be combined to obtain other probabilities. • What are the rules of probability?

17. Approaches to Probability Axiomatic Classical (A Priori) Frequency or Empirical (A Posteriori) Subjective

18. Axiomatic Approach Given a finite sample space S and an event A in S, we define P(A), the probability of A, to be a value of an additive set function P, which must satisfy the following three conditions: AXIOM 1. P(A)  0 for any event A in S. AXIOM 2. P(S) = 1

19. Axiomatic Approach AXIOM 3. If A1, A2 …, AK is a finite collection of mutually exclusive events in S, then

20. Classical Approach If an experiment can result in n equally likely and mutually exclusive ways, and if nA of these outcomes have the characteristic A, then the probability of the occurrence of A, denoted by P(A), is defined to be the fraction

21. Frequency of Empirical Approach If an experiment is repeated or conducted n times, and if a particular attribute A occurred times, then an estimate of the probability of the event A is defined as: Note that Remark: Probability can be interpreted as relative frequency in the long run.

22. Relative Frequency vs n 1 0 1 2 3 . . . n n = number of trials performed

23. Example An experiment consists of tossing a fair coin three times in sequence. What is the probability that 2 heads will occur?

24. Example Solution 1 2 3 Outcome Likelihood 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8 H HHH HHT HTH HTT THH THT TTH TTT 1/2 H 1/2 T 1/2 H 1/2 H 1/2 1/2 T 1/2 T 0 H 1/2 H 1/2 1/2 1/2 T T 1/2 H 1/2 T 1/2 T 1

25. Example Solution - Continued • Define the event A to be “2 heads occur” • Since the n=8 outcomes are equally likely and mutually exclusive, the condition for the classical approach to probability is satisfied. Therefore

26. Example An biased coin (likelihood of a head is 0.75) is tossed three times in sequence. What is the probability that 2 heads will occur?

27. Example Solution 1 2 3 Outcome likelihood 27/64 9/64 9/64 3/64 9/64 3/64 3/64 1/64 H HHH HHT HTH HTT THH THT TTH TTT 3/4 H 3/4 T 1/4 H 1/4 H 3/4 3/4 T 1/4 T 0 H 3/4 H 1/4 3/4 1/4 T T 1/4 H 3/4 T 1/4 T 1

28. Example Solution - Continued • Define the event A to be “2 heads occur” • The n=8 outcomes are mutually exclusive, but are not equally likely. • Therefore, the classical approach to probability is not applicable. P(A) = ?

29. Subjective Approach

30. Probability and Statistics Objectives • Model population to mathematically describe physical relationships • Describe variability • Quantify uncertainty • Analyze data to increase knowledge – draw conclusions and make decisions • Describe data numerically and graphically • Make inference about underlying population