1. Introduction to probability (1)

2. Probability is the language we use to model uncertainty, of the outcomes such as a scientific experiments or any action. These outcomes could have been different. Probability is underlying foundation on which the important methods of inferential statistics are built.

3. Introduction to probability (1) Example: Suppose that you were to win the top prize is a lotto five times, there would be investigations and accusations that you were some how cheating. This is incredibly; this is exactly how statisticians think. We reject luck as a reasonable explanation based on very low probabilities, so the statisticians’ use the rare event rule.

4. Introduction to probability (1) Rare Event Rule for Inferential Statistics If under a given assumption (such as a lottery being fair), the probability of a particular observed even (such as five lottery wins) is exactly small, we conclude that the assumption is probably not correct. In any experiment it produces outcomes and we should no some definitions to those outcomes:

5. Introduction to probability (1) Definitions: An Event: It is any collection of results or outcomes of procedures. Simple Event: It is an outcome or an event that can’t be further broken down into simpler components. Sample Space: The sample space for a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further. Sample space is represented by the symbol

6. Introduction to probability (1) Example: ProcedureExample of eventSample space Roll one die 5,4 [are simple event] {1, 2, 3, 4, 5, 6} Roll two dies 7(not a simple event) {1-1,1-2,…..,6-6}

7. Introduction to probability (1) Before we start in probability we should know about: 1.Data types: a.Discrete. b. Continuous. 2. Counting Techniques: a.Permutations (التباديل) b.Combinations (التوافيق).

8. Introduction to probability (1) Definitions: Permutation is an arrangement of objects in different orders. There are basically two types of permutation. 1. Repetition is allowed. 2. No repetition.

9. Introduction to probability (1) A) Permutation with Repetition: here order is important (الترتيب مهم والتكرار مسموح). If you have n things to choose from and you choose r of them then the permutations are:

10. Introduction to probability (1) Example: If there are 6 numbers {1, 2, 3, 4, 5, 6} and you choose 4 of them with repetition then we have permutations

11. Introduction to probability (1) B) Permutation without Repetition: here order is important (الترتيب مهم والتكرار مسموح) The formula of permutation without repetition is:

12. Introduction to probability (1) Example: Suppose we have a “ABCD” litters how many words we can compose from them with 4 litters without repetition.

13. Introduction to probability (1) ABCD ABCDBACDCABDDABC ABDCBADCCADBDACB ACBDBCADCBADDBAC ACDBBCDACBDADBCA ADBCBDACCDABDCAB ADCBBDCACDBADCBA

14. Introduction to probability (1) But how many words with 4 litters we can compose from 4 litters with repetition? Solution: Note: 0! = 1 1! = 1

15. Introduction to probability (1) Example: If A:={p, q, r, s} find the number of permutation for 3 elements: A) Without repetition

16. Introduction to probability (1) Example: If A:={p, q, r, s} find the number of permutation for 3 elements: B) With repetition