Online examination system Essay Example for Free

Online examination system Essay Symmetric key length, public-key key length, comparing symmetric and public-key key length, Key Management: (3 Lectures) Generating keys, transferring keys, verifying keys, using keys, updating keys, storing keys, compromised keys, lifetime of keys, destroying keys, public key management, Algorithm Types and Modes: (10 Lecture) Electronic codebook mode, block replay, cipher block chaining mode, stream ciphers, cipher-feedback mode, output feedback mode, choosing a cipher mode, block ciphers vs. tream ciphers, Choosing an Algorithm, Choosing an algorithm, public-key vs. symmetric cryptography, encrypting communication channels, encrypting data for storage, Security Models, Windows, Linux, Web, Cookies, Biometrics (gummy fingers), Tempest, Viruses, Firewalls, System/Comprehensive Security Analysis Text Book/s – Stallings, W and Brown, L. , Computer Security: principles and practice, 2nd edition, 2011 Reference Material – – William Stallings, Cryptography and Network Security, 4th Edition, 2005, ISBN-13: 978-0131873162 Kaufman, Perlman, Speciner, Network Security: Private Communication in a Public World, 2nd Edition, Prentice Hall, 2002, ISBN-13: 978-0130460196 – Other books will be used in the course if necessary. Department Of Computer Science Information Technology, HIC Jauharabad Campus Instructional Aids/Resources Assessment Criteria Class Assessment 35% Quizzes and Test 15 Assignment and Presentations 10 Attendance and Class Participations 5 Assignments + Quizzes Mid 25% If Required: Final 40% If Required: Total 100%.

Not a Problem :: Argumentative Internet Equal Access Essays

Not a Problem In today's hi-tech world, there are millions of people who are connected through the internet. Almost all of these users are living in industrialized nations, such as the United States. But there are many nations that lack the infrastructure necessary to support such forms of communication. Many of these nations are third world countries. In this essay, I am going to discuss the benefits and drawbacks of providing deprived regions of the world with such technology, and I will offer some better ways to spend our limited resources. I would have a difficult time arguing the case that no good could come from a world where everyone is connected to the internet. In fact, the world could benefit greatly if everyone was connected. The challenge lies in weighing the costs and the benefits. Is it worth the time and money needed to bring these people together? If general population of internet free individuals had the capabilities to connect to the rest of the world, would they utilize the technology? Do other issues need to be taken care of before we about giving everyone access to a computer? These are all things that need to be considered when addressing the topic of fair access. To get things started, how would giving people, who are currently without access to the internet, some form of connectivity help the rest of the world? I think that bringing the deprived people a tool as useful as the internet would do great things. Granting accessibility to the near endless expanse that we call cyberspace would bring a wealth of information to the destitute and uninformed. People would be able to communicate and express themselves in ways that had never been possible. They would be able to make educated decisions about issues that they may have otherwise not known about. The internet would bring another dimension to millions of people's lives. But how useful would this information be when more than nine hundred million adults cannot read, and millions more are considered functionally illiterate (Mooney 366). I will admit that bringing the internet to these people would bring them vast quantities of information, but that information is useless if they do not have the fundamental knowledge needed to decode the text that is the foundation of every webpage. Consider this for example: I could hand you a book containing all the secrets of the universe, but if the book was written in code, you would not be able to do anything with that knowledge.

Assessment Of Learning Essay

Assessment of learning refers to strategies designed to confirm what students know, demonstrate whether or not they have met curriculum outcomes or the goals of their individualized programs, or to certify proficiency and make decisions about students’ future programs or placements. It is designed to provide evidence of achievement to parents, other educators, the students themselves, and sometimes to outside groups (e. g. , employers, other educational institutions). Assessment of learning is the assessment that becomes public and results in statements or symbols about how well students are learning. It often  contributes to pivotal decisions that will affect students’ futures. It is important, then, that the underlying logic and measurement of assessment of learning be credible and defensible. TEACHERS’ ROLES IN ASSESSMENT OF LEARNING Because the consequences of assessment of learning are often far-reaching and affect students seriously, teachers have the responsibility of reporting student learning accurately and fairly, based on evidence obtained from a variety of contexts and applications. Effective assessment of learning requires that teachers provide †¢ a rationale for undertaking a particular assessment of learning at a particular  point in time †¢ clear descriptions of the intended learning †¢ processes that make it possible for students to demonstrate their competence and skill †¢ a range of alternative mechanisms for assessing the same outcomes †¢ public and defensible reference points for making judgements Rethinking Classroom Assessment with Purpose in Mind †¢ 55. The purpose of assessment that typically comes at the end of a course or unit of instruction is to determine the extent to which the instructional goals have been achieved and for grading or certification of student achievement. (Linn and Gronlund, Measurement and Assessment in Teaching ) Reflection: Think about an example of assessment of learning in your own teaching and try to develop it further as you read this chapter. 56 †¢ Rethinking Classroom Assessment with Purpose in Mind Chapter 5 †¢ transparent approaches to interpretation †¢ descriptions of the assessment process †¢ strategies for recourse in the event of disagreement about the decisions. With the help of their teachers, students can look forward to assessment of learning tasks as occasions to show their competence, as well as the depth and breadth of their learning. PLANNING ASSESSMENT OF LEARNING The purpose of assessment of learning is to measure, certify, and report the level  of students’ learning, so that reasonable decisions can be made about students. There are many potential users of the information:†¢ teachers (who can use the information to communicate with parents about their children’s proficiency and progress) †¢ parents and students (who can use the results for making educational and vocational decisions) †¢ potential employers and post-secondary institutions (who can use the information to make decisions about hiring or acceptance) †¢ principals, district or divisional administrators, and teachers (who can use the information to review and revise programming). Assessment of learning requires the collection and interpretation of information about students’ accomplishments in important curricular areas, in ways that represent the nature and complexity of the intended learning. Because genuine learning for understanding is much more than just recognition or recall of facts or algorithms, assessment of learning tasks need to enable students to show the complexity of their understanding. Students need to be able to apply key concepts, knowledge, skills, and attitudes in ways that are authentic and consistent with current thinking in the knowledge domain. What am I  assessing? Why am I assessing? Rethinking Classroom Assessment with Purpose in Mind †¢ 57 Assessment of Learning In assessment of learning, the methods chosen need to address the intended curriculum outcomes and the continuum of learning that is required to reach the outcomes. The methods must allow all students to show their understanding and produce sufficient information to support credible and defensible statements about the nature and quality of their learning, so that others can use the results in appropriate ways. Assessment of learning methods include not only tests and examinations, but  also a rich variety of products and demonstrations of learning—portfolios, exhibitions, performances, presentations, simulations, multimedia projects, and a variety of other written, oral, and visual methods (see Fig. 2. 2, Assessment Tool Kit, page 17). What assessment method should I use? Graduation Portfolios Graduation portfolios are a requirement for graduation from British Columbia and Yukon Senior Years schools. These portfolios comprise collections (electronic or printed) of evidence of students’ accomplishments at school, home, and in the community, including demonstrations of  their competence in skills that are not measured in examinations. Worth four credits toward graduation, the portfolios begin in Grade 10 and are completed by the end of Grade 12. The following are some goals of graduation portfolios: †¢ Students will adopt an active and reflective role in planning, managing, and assessing their learning. †¢ Students will demonstrate learning that complements intellectual development and course-based learning. †¢ Students will plan for successful transitions beyond Grade 12. Graduation portfolios are prepared at the school level and are based on specific Ministry criteria and standards. Students use the criteria and standards as guides for planning, collecting, and presenting their evidence, and for self-assessing. Teachers use the criteria and standards to assess student evidence and assign marks. There are three major components of a graduation portfolio: 1. Portfolio Core (30 percent of the mark). Students must complete requirements in the following six portfolio organizers: arts and design (respond to an art, performance, or design work); community involvement and responsibility (participate co-operatively and respectfully in a  service activity); education and career planning (complete a graduation transition plan); Employability skills (complete 30 hours of work or volunteer experience); information technology (use information technology skills); personal health (complete 80 hours of moderate to intense physical activity). 2. Portfolio Choice (50 percent of the mark). Students expand on the above areas, choosing additional evidence of their achievements. 3. Portfolio Presentation (20 percent of the mark). Students celebrate their learning and reflect at the end of the portfolio process. ( Portfolio Assessment and Focus Areas: A Program Guide) 58 †¢ Rethinking Classroom Assessment with Purpose in Mind Chapter 5 Assessment of learning needs to be very carefully constructed so that the information upon which decisions are made is of the highest quality. Assessment of learning is designed to be summative, and to produce defensible and accurate descriptions of student competence in relation to defined outcomes and, occasionally, in relation to other students’ assessment results. Certification of students’ proficiency should be based on a rigorous, reliable, valid, and equitable process of assessment and evaluation. Reliability  Reliability in assessment of learning depends on how accurate, consistent, fair, and free from bias and distortion the assessment is. Teachers might ask themselves: †¢ Do I have enough information about the learning of this particular student to make a definitive statement? †¢ Was the information collected in a way that gives all students an equal chance to show their learning? †¢ Would another teacher arrive at the same conclusion? †¢ Would I make the same decision if I considered this information at another time or in another way? Reference Points Typically, the reference points for assessment of learning are the learning  outcomes as identified in the curriculum that make up the course of study. Assessment tasks include measures of these learning outcomes, and a student’s performance is interpreted and reported in relation to these learning outcomes. In some situations where selection decisions need to be made for limited positions (e. g. , university entrance, scholarships, employment opportunities), assessment of learning results are used to rank students. In such norm-referenced situations, what is being measured needs to be clear, and the way it is being measured needs to be transparent to anyone who might use the assessment  results. Validity Because assessment of learning results in statements about students’ proficiency in wide areas of study, assessment of learning tasks must reflect the key knowledge, concepts, skills, and dispositions set out in the curriculum, and the statements and inferences that emerge must be upheld by the evidence collected. How can I ensure quality in this assessment process? Rethinking Classroom Assessment with Purpose in Mind †¢ 59 Assessment of Learning Record-Keeping Whichever approaches teachers choose for assessment of learning, it is their records that provide details about the quality of the measurement. Detailed records of the various components of the assessment of learning are essential, with a description of what each component measures, with what accuracy and against what criteria and reference points, and should include supporting evidence related to the outcomes as justification. When teachers keep records that are detailed and descriptive, they are in an excellent position to provide meaningful reports to parents and others. Merely a symbolic representation of a student’s accomplishments (e. g. , a letter grade or percentage) is inadequate. Reports to parents and others should identify the  intended learning that the report covers, the assessment methods used to gather the supporting information, and the criteria used to make the judgement. Feedback to Students Because assessment of learning comes most often at the end of a unit or learning cycle, feedback to students has a less obvious effect on student learning than assessment for learning and assessment as learning. Nevertheless, students do Ho w can I use the information from this assessment? Guidelines for Grading 1. Use curriculum learning outcomes or some clustering of these (e. g. , strands) as the basis for grading. 2.  Make sure that the meaning of grades comes from clear descriptions of curriculum outcomes and standards. If students achieve the outcome, they get the grade. (NO bell curves! ) 3. Base grades only on individual achievement of the targeted learning outcomes. Report effort, participation, and attitude, for example, separately, unless they are a stated curriculum outcome. Any penalties (e. g. , for late work, absences), if used, should not distort achievement or motivation. 4. Sample student performance using a variety of methods. Do not include all assessments in grades. Provide ongoing feedback on formative  performance using words, rubrics, or checklists, not grades. 5. Keep records in pencil so they can be updated easily to take into consideration more recent achievement. Provide second-chance assessment opportunities (or more). Students should receive the highest, most consistent mark, not an average mark for multiple opportunities. 6. Crunch numbers carefully, if at all. Consider using the median, mode, or statistical measures other than the mean. Weight components within the final grade to ensure that the intended importance is given to each learning outcome. 7. Make sure that each assessment meets quality standards (e.g. , there should be clear targets, clear purpose, appropriate target-method match, appropriate sampling, and absence of bias and distortion) and is properly recorded and maintained (e. g. , in portfolios, at conferences, on tracking sheets). 8. Discuss and involve students in grading at the beginning and throughout the teaching and learning process. (Adapted from O’Connor, How to Grade for Learning ) Resource: Marzano, Transforming Classroom Grading 60 †¢ Rethinking Classroom Assessment with Purpose in Mind Chapter 5 rely on their marks and on teachers’ comments as indicators of their level of  success, and to make decisions about their future learning endeavours. Differentiating Learning In assessment of learning, differentiation occurs in the assessment itself. It would make little sense to ask a near-sighted person to demonstrate driving proficiency without glasses. When the driver uses glasses, it is possible for the examiner to get an accurate picture of the driver’s ability, and to certify him or her as proficient. In much the same way, differentiation in assessment of learning requires that the necessary accommodations be in place that allow students to make the particular learning visible. Multiple forms of assessment offer multiple pathways for making student learning transparent to the teacher. A particular curriculum outcome requirement, such as an understanding of the social studies notion of conflict, for example, might be demonstrated through visual, oral, dramatic, or written representations. As long as writing were not an explicit component of the outcome, students who have difficulties with written language, for example, would then have the same opportunity to demonstrate their learning as other students. Although assessment of learning does not always lead teachers to differentiate  instruction or resources, it has a profound effect on the placement and promotion of students and, consequently, on the nature and differentiation of the future instruction and programming that students receive. Therefore, assessment results need to be accurate and detailed enough to allow for wise recommendations. Reporting There are many possible approaches to reporting student proficiency. Reporting assessment of learning needs to be appropriate for the audiences for whom it is intended, and should provide all of the information necessary for them to make reasoned decisions. Regardless of the form of the reporting,  however, it should be honest, fair, and provide sufficient detail and contextual information so that it can be clearly understood. Traditional reporting, which relies only on a student’s average score, provides little information about that student’s skill development or knowledge. One alternate mechanism, which recognizes many forms of success and provides a profile of a student’s level of performance on an emergent-proficient continuum, is the parent- student-teacher conference. This forum provides parents with a great deal of information, and reinforces students’ responsibility for their learning. The Communication System Continuum: From Symbols to Conversations (O’Connor, How to Grade for Learning ) Grades Report cards (grades and brief comments) Infrequent informal communications Parent-teacher interviews Report cards with expanded comments Frequent informal communication Student-involved conferencing Student-led conferencing Reflection: What forms do your reports of student proficiency take? How do these differ according to audience? Rethinking Classroom Assessment with Purpose in Mind †¢ 61 Assessment of Learning An Example of Assessment of Learning. Elijah was interested in assessing student mastery of both the modern and the traditional skills required for survival in the Nunavut environment where he teaches. The overarching theme of survival is taught in the early grades and culminates at the senior level in a course delivered in Inuktitut. Students learn how to take care of themselves and others, and how to adapt what they know to the situation at hand. Survival requires not only skills and knowledge, but also a concept the Inuit people call qumiutit, or the ability in an emergency situation to pull out of stored memory information that will enable a person to cope, not panic. Traditionally, this was learned in a holistic manner, grounded in Inuit traditional guiding principles that were nurtured and developed from birth, and taught and reinforced in daily living. Throughout the term, Elijah took his students to an outdoor area to practise on-the-land survival activities, using both traditional and modern methods. He always took with him a knowledgeable Elder who could give the students the information they needed to store away in case of emergency. The students watched demonstrations of a skill a number of times. Each student then practised on his or her own, as Elijah and the Elder observed and assisted. Elijah knew that students need to have a high level of expertise in the survival skills appropriate for the northern natural environment. Elijah assessed each student on each survival skill (e. g. , making fire the traditional way, tying the knots required for the qamutik cross-pieces on a sled). What am I assessing? I am assessing each student’s performance of traditional and modern survival skills. Why am I assessing? I want to know which survival skills each s tudent has mastered and their readiness to s urvive in the natural environment. 62 †¢ Rethinking Classroom Assessment with Purpose in Mind. Chapter 5 Elijah knew that the best way to determine if students have mastered the skills is to have them perform them. When students believed they were ready, Elijah created an opportunity for them to demonstrate the mastered skill to a group of Elders, who then (individually, then in consensus) determined if the performance was satisfactory. A student’s competence in a survival skill is often demonstrated by an end product. For example, competence in knot tying is demonstrated by a knot that serves its purpose, and competence in fire building is demonstrated by a fire that is robust. As the Elders judged each student’s performance of the skills, Elijah recorded the results. He shared the information with each student and his or her parents in a final report, as shown here. Ho w can I use the informatio n from this assessment? Now that I know which skills each of the students has mastered, I can report this information to the s tudents and their parents. I can use this information to identify a learning path for each s tudent. How can I ensure quality in this assessment process? Ensuring quality with this approach involves clear criteria: either the student performs the skill s uccessfully or does not. I need to provide adequate opportunities for the s tudent to demonstrate the skills under various conditions and at various times. What assessment method should I use? I need an approach in which students can demonstrate the traditional survival skills that they learned. The method I choose should also allow me to identify which skills they did not master. Rethinking Classroom Assessment with Purpose in Mind †¢ 63 Assessment of Learning Shelters: †¢ emergency shelters †¢ igloo building4 †¢ qamaq5 †¢ tents Transportation needs: †¢ making the knots required for the qamutik cross pieces on a sled †¢ building a kayak/umiak. †¢ fixing a snowmobile (spark plugs, repairing track, drive belt) †¢ keeping a boat seaworthy Navigational issues: †¢ reading the land †¢ reading the sky †¢ understanding seasonal variations †¢ reading inuksuit †¢ using GPS †¢ map reading Preparation for land travel: †¢ packing a qamutiq (sled) †¢ load, balance †¢ necessities: snow knife, rope, food, water, heat source †¢ letting others know where you are going †¢ necessary tools, supplies, snowmobile parts, fuel †¢ using communication devices Food sources: †¢ plants and their nutritional properties †¢ hunting, skinning, and cutting up seal, caribou, etc. †¢ kinds of food to take on the land,  and their nutritional properties ____________________ 4. Expertise in igloo building includes understanding of types of snow, the shape and fit of blocks, and the use of a snow- knife. 5. A qamaq is a rounded house, built of scrap wood or bones, and covered with skins, cardboard, or canvas. Report on Survival Skills Student: _______________________________________________ Date: _______________________ Traditional Survival Skills Modern Survival Skills Adaptability to the Seasons Attitude Success Next Steps 1) Skills Building a fire / means of keeping warm: †¢ fuel sources †¢ getting a spark †¢ propane heaters, stoves †¢ clothing. 2) Relationship to the Seasons Assessing conditions / recognizing danger signs: †¢ seasonal changes †¢ land changes †¢ water changes †¢ wind changes †¢ weather changes Climatic changes: †¢ weather changes and how this affects the land and water †¢ knowledge of animals and their characteristics and behaviours 3) Attitudinal Influences (Having the right attitude to learn) †¢ respect for the environment (cleaning up a campsite upon leaving, dealing with the remains of an animal, not over-hunting/fishing) †¢ respect for Elders and their knowledge †¢ ability to learn from Elders 64 †¢ Rethinking Classroom Assessment with Purpose in Mind Chapter 5. Elijah’s report identified which of the students had mastered the specified skills required to survive in the Nunavut environment. It outlined other areas (such as adaptability to the seasons and attitudinal influences) about which peers, parents, and family members would need to provide input before a comprehensive assessment could be made. The assessment also identified those students not yet ready to survive in the natural environment. But the Elders did not stop working with the students who did not reach mastery. Elders see learning as an individual path in which skills, knowledge, and attitudes are acquired along the way. If a particular skill was beyond the capability of a student, the Elders identified other areas where that person could contribute to the common good of the community, and was accepted for the gifts he or she brought to the group. In this way, the Elders helped Elijah differentiate the learning path for each of his students. SUMMARY OF PLANNING ASSESSMENT OF LEARNING Assessment for Learning Assessment as Learning Assessment of Learning Why Assess? To enable teachers to determine next steps in advancing student learning to guide and provide opportunities for each student to monitor and critically reflect on his or her learning, and  identify next steps Assess What? each student’s progress and learning needs in relation to the curricular outcomes each student’s thinking about his or her learning, what strategies he or she uses to support or challenge that learning, and the mechanisms he or she uses to adjust and advance his or her learning. What Methods? a range of methods in different modes that make students’ skills and understanding visible a range of methods in different modes that elicit students’ learning and metacognitive processes Ensuring Quality †¢ accuracy and consistency of observations and interpretations of student learning  Ã¢â‚¬ ¢ clear, detailed learning expectations †¢ accurate, detailed notes for descriptive feedback to each student †¢ accuracy and consistency of student’s self-reflection, self-monitoring, and self-adjustment. †¢ engagement of the student in considering and challenging his or her thinking †¢ students record their own learning Using the Information †¢ provide each student with accurate descriptive feedback to further his or her learning †¢ differentiate instruction by continually checking where each student is in relation to the curricular outcomes †¢ provide parents or guardians with descriptive feedback about student  learning and ideas for support. †¢ provide each student with accurate descriptive feedback that will help him or her develop independent learning habits †¢ have each student focus on the task and his or her learning (not on getting the right answer) †¢ provide each student with ideas for adjusting, rethinking, and articulating his or her learning †¢ provide the conditions for the teacher and student to discuss alternatives †¢ students report about their learning Rethinking Classroom Assessment with Purpose in Mind. †¢ 65 Assessment of Learning to certify or inform parents or others of student’s proficiency in  relation to curriculum learning outcomes the extent to which students can apply the key concepts, knowledge, skills, and attitudes related to the curricular outcomes a range of methods in different modes that assess both product and process †¢ accuracy, consistency, and fairness of judgements based on high-quality information †¢ clear, detailed learning expectations †¢ fair and accurate summative reporting †¢ indicate each student’s level of learning †¢ provide the foundation for discussions on placement or promotion †¢ report fair, accurate, and detailed information that can be used to decide the next steps in a student’s learning.   

What Are Probability Axioms

One strategy in mathematics is to start with a few statements, then build up more mathematics from these statements. The beginning statements are known as axioms. An axiom is typically something that is mathematically self-evident. From a relatively short list of axioms, deductive logic is used to prove other statements, called theorems or propositions. The area of mathematics known as probability is no different. Probability can be reduced to three axioms. This was first done by the mathematician Andrei Kolmogorov. The handful of axioms that are underlying probability can be used to deduce all sorts of results. But what are these probability axioms? Definitions and Preliminaries In order to understand the axioms for probability, we must first discuss some basic definitions. We suppose that we have a set of outcomes called the sample space S.  This sample space can be thought of as the universal set for the situation that we are studying. The sample space is comprised of subsets called events E1, E2, . . ., En.   We also assume that there is a way of assigning a probability to any event E. This can be thought of as a function that has a set for an input, and a real number as an output. The probability of the event E is denoted by P(E). Axiom One The first axiom of probability is that the probability of any event is a nonnegative real number. This means that the smallest that a probability can ever be is zero and that it cannot be infinite. The set of numbers that we may use are real numbers. This refers to both rational numbers, also known as fractions, and irrational numbers that cannot be written as fractions. One thing to note is that this axiom says nothing about how large the probability of an event can be. The axiom does eliminate the possibility of negative probabilities. It reflects the notion that smallest probability, reserved for impossible events, is zero. Axiom Two The second axiom of probability is that the probability of the entire sample space is one. Symbolically we write P(S) 1. Implicit in this axiom is the notion that the sample space is everything possible for our probability experiment and that there are no events outside of the sample space. By itself, this axiom does not set an upper limit on the probabilities of events that are not the entire sample space. It does reflect that something with absolute certainty has a probability of 100%. Axiom Three The third axiom of probability deals with mutually exclusive events. If E1 and E2 are mutually exclusive, meaning that they have an empty intersection and we use U to denote the union, then P(E1 U E2 ) P(E1) P(E2). The axiom actually covers the situation with several (even countably infinite) events, every pair of which are mutually exclusive. As long as this occurs, the probability of the union of the events is the same as the sum of the probabilities: P(E1 U E2 U . . . U En ) P(E1) P(E2) . . . En Although this third axiom might not appear that useful, we will see that combined with the other two axioms it is quite powerful indeed. Axiom Applications The three axioms set an upper bound for the probability of any event. We denote the complement of the event E by EC. From set theory, E and EC have an empty intersection and are mutually exclusive. Furthermore E U EC S, the entire sample space. These facts, combined with the axioms give us: 1 P(S) P(E U EC) P(E) P(EC) . We rearrange the above equation and see that P(E) 1 - P(EC). Since we know that probabilities must be nonnegative, we now have that an upper bound for the probability of any event is 1. By rearranging the formula again we have P(EC) 1 - P(E). We also can deduce from this formula that the probability of an event not occurring is one minus the probability that it does occur. The above equation also provides us a way to calculate the probability of the impossible event, denoted by the empty set. To see this, recall that the empty set is the complement of the universal set, in this case SC. Since 1 P(S) P(SC) 1 P(SC), by algebra we have P(SC) 0. Further Applications The above are just a couple of examples of properties that can be proved directly from the axioms. There are many more results in probability. But all of these theorems are logical extensions from the three axioms of probability.