STT When trying to evaluate the likelihood of random events we are using following wording.

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Introduction to Chapter 2. Probability. When trying to evaluate the likelihood of random events we are using following wording. Provide your own corresponding examples. Subjective probability.

1 Introduction to Chapter 2. Probability. When trying to evaluate the likelihood of random events we are using following wording. Provide your own corresponding examples. Subjective probability. An individual person's measure of belief that an event will occur. Example. It is makes perfectly good sense intuitively to talk about the probability that the Dow Jones average will go up tomorrow. Downside of subjective probability. Since it is subjective, one person's probability (e.g., that the Dow Jones will go up tomorrow) may differ from another's. The subjective perspective of probability fits well with Bayesian statistics, which are an alternative to the more common frequentist statistical methods. Internet sources of visualization of random experiments

Empirical definition of probability. Intuitively, probability of an event is a long-run proportion of times the event occurs in in many independent repetitions of the experiment.

2 Empirical definition of probability. Intuitively, probability of an event is a long-run proportion of times the event occurs in in many independent repetitions of the experiment. Law of Large Numbers: # of times A hasoccured # of repetitions P( A) Empirical probability is based on experiments. You physically perform experiments and calculate the odds from the results of performed experiment. Internet sources of visualization of random experiments. The empirical view of probability is the one that is used in most statistical inference procedures. The researchers considered so-called frequentist statistics the numerical characteristics of the sample. The empirical (frequentist) view is what gives credibility to conclusions about the population based on sampling. Example. If we randomly choose a sample of 1000 students from the population of all 45,000 students enrolled in MSU, then the average of college expenses for the sample is a reasonable estimate of the average for the population. Randomness doesn t mean Chaos. Random evens and processes follow specific statistical laws. 2

3 Chapter 2. Probability The probability theory is a study about randomness and uncertainty. Objectives. Experiment with random outcomes. Sample space. Random events. Axioms of probability. Properties of probability. Experiment with equally likely outcomes. Classical definition of probability. Introduction to sets theory. Introduction to combinatorial theory. Counting techniques Basic Definitions (terminology) and Notations Random experiment Sample space (for a random experiment) a process whose outcome cannot be predicted the set of all possible outcomes of the experiment Common notation: S, Size of the sample space n Example Random experiment toss a coin twice Outcomes Sample space Size of the sample space 3

4 Event Example Random experiment toss a coin twice Sample space HH, HT, TH, TT Event: {getting a tail} Opposite, complement event getting no tails Mathematical Definition of Event An event is a subset of the sample space. An event is said to occur if any of its outcomes occurs Clarification of Terminology An outcome is a single, simple, elementary result of an experiment The sample space is the set of all possible outcomes. A sample space is a set. An event is a subset of the sample space. An event is a set Number of Subsets (Events) in a Sample Space. If a sample space has n elements, then Examples of Sample there Spaces are and subsets Events. (events) 4

5 Example For the following random experiments describe the sample space and some events. Experiment Sample space Events Rolling a die Taking two Finals at MSU Checking quality of three resisters (standard/defective) How much cash every one of you have in your pocket Some Relations from Set Theory Name and Notation Description Complement event A A The set of all outcomes in that are not in A Union A B The set of outcomes that belong to at least one of the events Intersection A B The set of outcomes that are in both, A and B Difference A B The set of outcomes that are in A, but are not in B 5

exclusive or disjoint events. Example 2.1.

6 The Venn Diagram Impossible Event and Mutually Exclusive (Disjoint) Events. Definitions. is the notation for null (impossible) event event consisting of no outcomes When A B, A and B are said to be mutually exclusive or disjoint events. Example Shade the regions that correspond with A B C; A B C; A BC. ; 6

7 Example Which of the events A, B, and C are mutually exclusive? A in the shipment all items are standard B in the shipment is one defective item C in the shipment is at least one defective item Answer. A and B, A and C 2.2. Axioms, Interpretation, and Properties of Probability Classical Definition of Probability. Classical definition of probability is historically first model and related to counting outcomes of games (development of scientific approach started in 17th century by famous French scientists Pascal, Fermat, Laplace). 1. In a sample space of n equally likely outcomes the probability of any given Consider single a outcome random experiment is 1 and its sample space. Our goal is to assign to n. each event A a number P A that will measure the chances 2. Probability of the occurrence of any event of A the is event equal A. to sum of probabilities of outcomes that form this event. If the number of outcomes that formed A is m, then m P A n Generalization - finite number of outcomes, but without equally likely assumption. If a sample space has outcomes and an event A consists of n outcomes with assigned probability i p, then the probability sum of probabilities of outcomes that form the event A. n P P A i i P A A can be determined as 7

Example 2.2.3. Find the probability of getting even sum when rolling two fair dice. Solution.

n 36 m 6 2 4 2 2 18 P even sum 18 1 36 2 Internet sources of visualization of random experiments.

Consider a random experiment and its sample space.

8 Example Find the probability of getting even sum when rolling two fair dice. Solution. 36 outcomes are equally likely. n 36 m P even sum Internet sources of visualization of random experiments Axiomatic Definition of Probability. Consider a random experiment and its sample space. P A that will measure the Our goal is to assign to each event A a number chances of the occurrence of the event A. Axioms of Probability A, P A 0 Axiom1. For any event Axiom 2. P 1 Axiom 3. If A1, A2, A 3,... is an infinite collection of disjoint events, then P A1 A2 A3... P Ai i1 8

9 Remark P More Properties of Probability. For any event A, P A P A 1 P A 1 P A For any two events A and B, P A B P A PB P A B For any three events A, B, and C P A B P A C PB C P A B C P A B C P A P B P C Venn Diagrams Illustration of the Rule for Computing P A B P AB C and 9

Example 2.2.8. (#26 p.64 textbook) A certain system can experience three different types of defects. Let Ai i 1,2,3 denote the event that the system has a defect of type i. Suppose that P A 0.

88 b) What is the probability that the system has both type 1and type 2 defects? 1 2 1 1 1 2 1 2 1 1 1 2 A 0.13 P A A P A P A P A A P A A P A P A P A A P A1 2 0.12 0.07 0.

10 Example (#26 p.64 textbook) A certain system can experience three different types of defects. Let Ai i 1,2,3 denote the event that the system has a defect of type i. Suppose that P A 0.12 P A 0.07 P A P A A 0.13 P A A 0.14 P A A 0.10 P A1 A2 A a) What is the probability that the system does not have a type 1 defect? P A 1 P A P A b) What is the probability that the system has both type 1and type 2 defects? A 0.13 P A A P A P A P A A P A A P A P A P A A P A c) What is the probability that the system has both type 1and type 2 defects but not a type 3 defect? P A A A P A A P A A A P A A A d) What is the probability that the system has at most two of these defects? At most two is complement of all three. P at most two 1 P A A A

Recommendation. When you are working with this sort of problem, it is very helpful to draw corresponding Venn diagram and put the probabilities that correspond to all disjoint parts. 2.3.

11 Recommendation. When you are working with this sort of problem, it is very helpful to draw corresponding Venn diagram and put the probabilities that correspond to all disjoint parts Permutations and Combinations. To count classical probability, we need to know some combinatorics Definitions. An ordered subset is called a permutation. The number of permutation of size k that can be formed from the group of n n! elements is Pkn, n k! Example A monkey is playing with small wooded cubes with letters D, L, O, R, W. What is the probability that the monkey makes the word world? Answer ! 120 An unordered subset is called a combination. The number of combinations of size k that can be formed from the group of n elements is n! Ckn, k! n k! Pkn, Mention, that C kn, k! All these formulas are obtained by using multiplication rule for independent choice of a pair of objects: If for an element x we have n choices and for another element y m choices, then there is n m different ordered pairs, if the elements are chosen independently. 11

Example 2.3.3. A security lock opens on stored 5 letters sequence. What is the probability to open the lock by using an arbitrary sequence of 5 letters if a) digits can be repeated? 1 1 Answer.

12 Example A security lock opens on stored 5 letters sequence. What is the probability to open the lock by using an arbitrary sequence of 5 letters if a) digits can be repeated? 1 1 Answer b) letters must be all different? Answer Example A sample of five rivets is taking from the box with a 100 for quality control. Statistics shows that in average the machine produces 4% detective rivets. What is the probability that in the sample taking randomly for control will be 1) One defective rivet? 2) Two defective rivets? 3) At least one defective rivet? 4) Only standard rivets? 5) Are events 1) and 3) mutually exclusive? This problem is a case of sampling from the population with two types of elements, with N elements of type and M elements of another type. We will need to find the probability that in the sample will be n elements of first type and m elements of second type. Solution. C p C n, N m, M Cm n, N M 12

13 In the Example 2.3.4: Two types of elements of elements are standard and defective rivets. N 96 - number of standard rivets, M 4 - number of defective rivets Sample size nm 5. 1) n4, m 1 P one defective in the sample of 5 C C C 4, 96 1, 4 5,100 96! 4 4!92! 100! 5!95! By using the Properties of Factorials, expression for probability can be seriously simplified. Also, scientific calculators can be used for computing number of combinations. Idea of simplification expressions with factorials. 100! !... 5! 95! 5! 95! 2) n3, m 2 and computations are the same as in 1). C3, 96 C2, 4 Ptwo defective in the sampleof 5 C 3) The event {at least one defective} is the complement to the event {no defective}. The probability P{no defective}is much more easy to find than the probability P{at least one defective}. The event {no defective rivets in the sample} means that all five rivets were selected from 96 standard items. C5, 96 Thus, Pat least one defective 1 Pno defective 1 C 4) Pno defective C C 5, 96 5,100 5,100 5) No. {one defective} is a part of at least one defective}. 5,100 13

14 Example (modified #41p. 72 textbook). An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2,... 8, or 9, in succession. a) How many different possible PINs are there if there are no restrictions on the choice of digits? 4 Answer. 10 b) According to a representative of a local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited: b1) all four digits identical There are 10 PINs with all digits the same. b2) sequences of consecutive ascending or descending digits, such as 6543 There possible 7 4-digits sequences with ascending order: 0123, 1234, 2345, 3456, 4567, 5678, 6789 and 7 in descending order So, total number prohibited sequences b2) is 14. b3) any sequence starting with 19 (birth years are too easy to guess). There are 100 sequences starting with 19..: 1900, So, if one of the PINs in a) is randomly selected, what is the probability that it will be a legitimate PIN (that is, not be one of the prohibited sequences)? Total number of prohibited sequences is =124. So, the number of legitimate sequences is = Prandomly selected is legitimte one c) Someone has stolen an ATM card and knows that the first and last digits of the PIN are 8 and 1, respectively. What is the probability that the individual gains access to the account in the first try? The individual knows about the restrictions described in (b), so selects only from the legitimate possibilities. 14

15 There are 100 PINs in the form (8 - -1) and all of them are legitimate. 1 P gain access 100 d) Re-calculate the probability in c) if the first and last digits are 1 and 1, respectively. Eleven PINs in the form ( ) are prohibited: ( ) and the in the form (1 9 1) 10 more. So, there are 89 legitimate sequences in the form ( ) 1 P gain access in one try

I - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability

I - Probability. What is Probability? the chance of an event occuring. 1classical probability. 2empirical probability. 3subjective probability What is Probability? the chance of an event occuring eg 1classical probability 2empirical probability 3subjective probability Section 2 - Probability (1) Probability - Terminology random (probability)