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10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations
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10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations

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10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations

10.15.2013 Prj & Port Mgmt SftDev - Investment Analyzer - Projection and Simulations

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  • 1. IT and Engineering Portfolio & Strategy Management AAO-CO202: Jim Densmore Executive IT Specialist IT Architect and Technical Sales Specialist Rational Brand, SWG 23 October 2013 © 2013 IBM Corporation
  • 2. IBM 2013 – Project and Portfolio Management in Systems and Software Development Rational Focal Point Investment Analysis Component Projection and Simulations 15 October 2013 Jim Densmore Murray Cantor, Ph.D. Sr. Certified IT Specialist IBM Rational Software jdensmore@us.ibm.com Distinguished Engineer IBM Rational Software mcantor@us.ibm.com © 2013 IBM Corporation © 2013 IBM Corporation 2
  • 3. IBM Objectives  Motivate and understand an ROI-based method for you and your clients to align the development organization (IT, Engineering, etc.) with the business  See how IBM Rational provides a mechanism for clients to do this more easily – Process – Tooling – Use cases © 2013 IBM Corporation 3
  • 4. IBM Agenda • Business Challenges • Aligning IT or Engineering with the Business • An ROI based Solution • Focal Point and its Investment Analysis Component • Use Cases for Investment Analysis © 2013 IBM Corporation 4
  • 5. IBM Agenda • Business Challenges • Aligning IT or Engineering with the Business • An ROI based Solution • Focal Point and its Investment Analysis Component • Use Cases for Investment Analysis © 2013 IBM Corporation 5
  • 6. IBM Business Challenges Zero-growth budgets 2011 Demand for IT Services Do More! “The economic downturn forced deep cuts in IT budgets. Now, as CIO’s plan for the recovery, they are facing unprecedented demand for IT services from the business. At the same time, organizations are still keeping spending tightly under control.” CIOUpdate.com August 3, 2010 http://www.cioupdate.com/budgets/article.php/3896646/How-to-Get-the-Budget-You-Need-in-2011.htm Executive Programs CIO Survey Increases in CIO Budgets Over Previous Year, 1998-2009 (Worldwide) 18.0% 16.0% With Less! 15.0% 15.9% 14.0% 12.0% 9.7% 10.1% 10.0% 8.0% 6.0% 4.0% 1.3% 2.0% 0.0% Source: Gartner 1.6% 2.5% 2.7% 3.0% 3.2% 0.16% 0.0% 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Trend Line © 2013 IBM Corporation 6 6
  • 7. IBM Business Response Must (re!)balance resource allocations to support business innovation IT Spend 1/3 2/3  Often, ongoing operations & maintenance consume > 2/3 of budgets  This may leave too little for innovation Original graphic created by David Puzas, WW Marketing Executive for IBM Enterprise Services, 2007 © 2013 IBM Corporation 7 7
  • 8. IBM Agenda • Business Challenges • Aligning IT or Engineering with the Business • An ROI based Solution • Focal Point and its Investment Analysis Component • Use Cases for Investment Analysis © 2013 IBM Corporation 8
  • 9. IBM Alignment of IT (or Engineering) & Business Value is Crucial A disciplined, strategic translation is key to this alignment • Translation management and maintenance? • What is their common denominator? What challenges must we meet? • Business values evolve • The mapping to capabilities changes We can’t fund both projects. Which one do we keep? • Capabilities are always in flux © 2013 IBM Corporation 9
  • 10. IBM Value The conventional wisdom for estimating the worth of an incomplete development effort (project) provides little insight Conventional wisdom: no value until shipped 0 6 Ship date ? ? ? ?  Fails to acknowledge work already done  What scale do you use?  Provides no opportunity for ongoing value management  How does one avoid gaming? ?  Have you ever killed a project? How? © 2013 IBM Corporation 10
  • 11. IBM Agenda • Business Challenges • Aligning IT or Engineering with the Business • An ROI based Solution • Focal Point and its Investment Analysis Component • Use Cases for Investment Analysis © 2013 IBM Corporation 11
  • 12. IBM Bringing IBM’s math department to bear • Different activities  different analytics & productivity measures • Improve odds of delivery and predictability across the lifecycle Variance in cost, schedule… Inception Elaboration Construction Transition/Maintenance Industry spend ROI New platform New capability on existing platform High Variance Maintenance and small change requests Medium Variance Low Variance Time © 2013 IBM Corporation 12
  • 13. IBM This led to the approach: what might someone pay for it?  Imagine selling your incomplete development program – Buyer would spend now/today – Obtains the option to invest in its completion – Once completed, receives hoped-for benefits – So … buy the option to spend to receive uncertain benefits • This is like a call option with an uncertain strike price! • No wonder ROI can be difficult to measure  Reason about a fair price using incomplete market reasoning:  Expected cost to complete over time,  Expected value received over time,  The estimated risk (uncertainty) over time © 2013 IBM Corporation 13
  • 14. IBM Example  For what might Airbus sell their A350 XWB program? – In testing with reportedly good results – Maiden flight completed (June 2013) – Many orders at varying levels of firmness • 682 a/c ordered by 33 global customers • As of July 2013 (wikipedia)  Certainly not zero!  How much value do you see in your portfolio of incomplete projects? Does the value grow over time? ? ? ? ? ?  (What’s wrong with this image? Why?) © 2013 IBM Corporation 14
  • 15. IBM Agenda • Business Challenges • Aligning IT or Engineering with the Business • An ROI based Solution • Focal Point and its Investment Analysis Component • Use Cases for Investment Analysis © 2013 IBM Corporation 15
  • 16. IBM Rational Focal Point Hardwiring the linkage between strategy and execution • The Rational Team Concert integration allows users to prioritize and manage project scope and rollup project status • The System Architect integration connects the Enterprise Architecture perspective to the portfolio management perspective in Focal Point • FP’s Investment Analysis component assists users with financial modeling and business case assessment • Users can take advantage of advanced resource management allowing skill-based supply and demand tracking and balancing • Configuration templates included in the product helps users get up and running quickly © 2013 IBM Corporation 16
  • 17. IBM The best common scale of value is “Money” Investment Analysis provides financial measures for everyday use © 2013 IBM Corporation 17
  • 18. IBM IA presents streams with quarter-by-quarter values Explicit values (others are interpolated) © 2013 IBM Corporation 18
  • 19. IBM Each input assumption is found using simple input estimates  A triangular distribution is used – Commonly used in business decision making – Values are simple to understand – (Remember, the area of the triangle is 1 by definition) 0 L L, the lowest monetary value you believe could occur (no chance of a lower value) E H E, the most likely or expected monetary value H, the highest monetary value you believe could occur (no chance of a higher value) © 2013 IBM Corporation 19
  • 20. IBM Vignette: How long will it take?  You’re a development manager  You assign your developer a task  “How long will it take?” He hems & haws …  “Oh … about 20 weeks!”  The developer returns to his cube  His buddy prairie dogs & says, “Hey. Did you get the job?” “Yeah.” “How long will it take?” “Oh, about 8 weeks!”  … What happened? © 2013 IBM Corporation 20
  • 21. IBM Vignette: How long will it take?  You’re a development manager Conclusion: ask for all three numbers! The  You assign your developer a task three-number (triangular distribution) answer is both more honest as well as more proactive;  “How long will it take?” He hems & haws … it changes the game  “Oh … about 20 weeks!”  The developer returns to his cube  His buddy prairie dogs & says, “Hey. Did you get the job?” “Yeah.” “How long will it take?” “Oh, about 8 weeks!” 0 8 12 20  … What happened? © 2013 IBM Corporation 21
  • 22. IBM IA represents uncertainty with 3 values for each quarter (period) © 2013 IBM Corporation 22
  • 23. IBM You assign a discount rate on each (stakeholder) page Discount rate determination is an entire, important subject on its own:  It should never be zero (the default, by the way)  Should costs and benefits have the same or different rate?  Too high and long-term investment is discouraged  Too low? Might be worth doing, but mañana is soon enough! ? ? ? ?  How do you handle discount rates? © 2013 IBM Corporation ? 23
  • 24. IBM A view (page) can be developed for each Stakeholder Revenue Devel. Costs Maint. Costs Combined © 2013 IBM Corporation 24
  • 25. IBM Monte Carlo analysis permits the needed arithmetic operations on random variables  Run thousands of simulations of the investment value (the NPV) – Each time a value is picked from each triangularly distributed random variable – Each value chosen is based on the likelihood of that value occurring, according to the distribution of the associated random variable  For each simulation, the investment value is calculated  Finally, we build up the histogram of investment values to obtain its distribution © 2013 IBM Corporation 25
  • 26. IBM A practical demonstration of the value of risk management Revenue Devel. Costs Maint. Costs Combined What is the effect of narrowing cost variance? © 2013 IBM Corporation 26
  • 27. IBM This shows Monte Carlo analysis demonstrating empirically the value of narrowing uncertainties in cost © 2013 IBM Corporation 27
  • 28. IBM Thus, our model for estimating NPV and ROI has some useful properties  Investment value varies continuously in time  Investment value improves when we invest: – In improving likelihood of delivery (reducing uncertainty in costs) – In improving the range of value, e.g. building reuse into solutions (that is, increasing the upside variance of benefits) Value Investment Value 0 Conventional Wisdom 6 Ship date © 2013 IBM Corporation 28
  • 29. IBM Bubble chart: standard way to represent a project portfolio “better” Strategic Value Radius = Cost 7 5 2 4 3 5 4 3 Project 6 8 6 $ 1 9 2 1 0 1 0 Traditionally, value & risk xx are based mostly on perceptions, opinions Risk 5 “worse”  Doesn’t show that incomplete projects have value  Creates a discontinuity in thinking  Rarely accounts for timing and total costs of ownership © 2013 IBM Corporation 29
  • 30. IBM This Investment Value Model puts monetary number$ on value and risk Investment xValue = Mean 2000 1500 Project 10 1000 Project 3 $ Value Project 2 2 500 5 3 Project 4 4 Project 5 0 1 Project 1 7 Project 6 8 6 Project 7 9 Project 9 -500 Project 8 10 -1000 0 Standard Deviation ? Normalized Risk 1 IV = Mean Normalized Risk = (Scaled) Standard Deviation © 2013 IBM Corporation 30
  • 31. IBM Agenda • Business Challenges • Aligning IT or Engineering with the Business • An ROI based Solution • Focal Point and its Investment Analysis Component • Use Cases for Investment Analysis © 2013 IBM Corporation 31
  • 32. IBM These two key questions support value-based decision making  How do you compare routine and innovative efforts to each other? ? ? ? ?  How do you manage project risk? ?  How do you motivate architectural robustness and reuse? Are we creating value? Is this program worth continuing? Current value, ROI to date Program onset: T0 Likely value at delivery, & likely ROI at delivery Today: T1 Program delivery: Td Management Decisions Supported: Monitoring Investment  Is program healthy?  Is program still needed?  Intervene?  Should we adjust content?  Cut losses?  Should we continue to invest? © 2013 IBM Corporation 32
  • 33. IBM The Model permits more objective management of the portfolio in the usual resource-constrained environment $ Value $5k 7 $4k 2 3 5 4 $3 k Legend Project 6 8   1 9 Questionable  6 Keep Cut? $2k 1 0 $1k 0 Normalized Risk 1 © 2013 IBM Corporation 33
  • 34. IBM The model can help choose among build scenarios Scenario A – Cobbled together, quick to field, minimum investment $ Value Scenario B – More reusable, extensible, lower O&M costs B $5k $4k A $3 k $2k $1k 0 Normalized Risk © 2013 IBM Corporation 1 34
  • 35. IBM Another use is to track improvement in value, and reduction of risk, throughout the project lifecycle  T1 is project onset; T2 and T3 are later times in the lifecycle $ Value  Movement from lower right to upper left shows that the investment (development) is delivering value T3 $5k $4k T2 $3 k T1 $2k $1k 0 Normalized Risk © 2013 IBM Corporation 1 35
  • 36. IBM IA is a key part of Delivering on these Five Keys to Success Blockers Its difficult to link an IT project portfolio to the Its difficult to link an IT project portfolio to the realization of business benefits realization of business benefits Five keys to success 1. Understand the value of investments and tradeoffs on content rather than becoming “bogged down” in project or resource detail Organizations do not have the ability to Organizations do not have the ability to reconcile both project delivery and reconcile both project delivery and architectural perspectives architectural perspectives 2. Build a stronger connection between customers, Enterprise Architecture and portfolio management disciplines to enable a broader viewpoint for decisions Organizations want to manage lifecycles, not Organizations want to manage lifecycles, not just projects just projects 3. Leverage lifecycle management of products, services or IT applications with customer, competitive and capability viewpoints that show how projects deliver against the lifecycle Project managers report on the status of a Project managers report on the status of a project, but this does not help improve project, but this does not help improve delivery capabilities within the organization delivery capabilities within the organization Project management tools focus on the needs Project management tools focus on the needs of the project manager helping them manage of the project manager helping them manage schedules, costs and resources but creates schedules, costs and resources but creates overhead for practitioners overhead for practitioners 4. Gain insight into how delivery practices impact project success, to inform process and skill improvements and organizational maturity 5. Provide teams with a collaborative platform that allows software delivery projects to become more automated, transparent and predictable across all disciplines © 2013 IBM Corporation 36
  • 37. IBM AAO-CO202 © 2013 IBM Corporation 37
  • 38. IBM AAO-CO202 www.ibm.com/software/rational © Copyright IBM Corporation 2013. All rights reserved. The information contained in these materials is provided for informational purposes only, and is provided AS IS without warranty of any kind, express or implied. IBM shall not be responsible for any damages arising out of the use of, or otherwise related to, these materials. Nothing contained in these materials is intended to, nor shall have the effect of, creating any warranties or representations from IBM or its suppliers or licensors, or altering the terms and conditions of the applicable license agreement governing the use of IBM software. References in these materials to IBM products, programs, or services do not imply that they will be available in all countries in which IBM operates. Product release dates and/or capabilities referenced in these materials may change at any time at IBM’s sole discretion based on market opportunities or other factors, and are not intended to be a commitment to future product or feature availability in any way. IBM, the IBM logo, Rational, the Rational logo, Telelogic, the Telelogic logo, and other IBM products and services are trademarks of the International Business Machines Corporation, in the United States, other countries or both. Other company, product, or service names may be trademarks or service marks of others. © 2013 IBM Corporation 38
  • 39. IBM For more interaction …  Join and collaborate on the TLE Community. – TLE Community: http://w3.ibm.com/connections/communities/service/html/communityview?co  TLE events are being scheduled continuously, so check the website frequently. – TLE Website: http://tle.atlanta.ibm.com/home.html © 2013 IBM Corporation 39
  • 40. IBM Alignment is Difficult • Concerns flow down the organization while measures (and data) flow up • We need tools to plan, track, and deliver on our commitments at every level Concerns Senior Manager commits to measures commits to measures Project Manager or Team Lead commits to measures Profit, Internal Rate of Return Delivery of business value through the optimal use of resources Measures Line of Business Executive Project deliverables, cost and schedule © 2013 IBM Corporation 40
  • 41. IBM Motivators and Blockers Line of Business Executive Senior Manager Project Manager or Team Lead Need more effective support for strategic decision making Linking IT project portfolio to Linking IT project portfolio to business benefit realization business benefit realization Reconciling project delivery Reconciling project delivery & architectural perspectives & architectural perspectives Want to manage lifecycles, Want to manage lifecycles, not just projects not just projects Need better metrics & analytics for continuous process improvement PMs’ status reports don’t PMs’ status reports don’t improve delivery capabilities improve delivery capabilities Must connect better with practitioners, enhance production PM tools focus on PM PM tools focus on PM needs but increase needs but increase overhead for practitioners overhead for practitioners © 2013 IBM Corporation 41
  • 42. Monte Carlo Simulation (Wikipedia IBM 1 has a good article at http://en.wikipedia.org/wiki/Monte_Carlo_method) Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to Monte Carlo methods (or Monte Carlo experiments) are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in simulating physical and mathematical systems. These methods are most compute their results. Monte Carlo methods are often used in simulating physical and mathematical systems. These methods are most suited to calculation by a computer and tend to be used when it is infeasible to compute an exact result with a deterministic algorithm. suited to calculation by a computer and tend to be used when it is infeasible to compute an exact result with a deterministic algorithm. This method is also used to complement the theoretical derivations. This method is also used to complement the theoretical derivations.  Monte Carlo methods are especially useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered  Monte Carlo methods are especially useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model). They are used to model phenomena with significant materials, strongly coupled solids, and cellular structures (see cellular Potts model). They are used to model phenomena with significant uncertainty in inputs, such as the calculation of risk in business. They are widely used in mathematics, for example to evaluate uncertainty in inputs, such as the calculation of risk in business. They are widely used in mathematics, for example to evaluate multidimensional definite integrals with complicated boundary conditions. When Monte Carlo simulations have been applied in space multidimensional definite integrals with complicated boundary conditions. When Monte Carlo simulations have been applied in space exploration and oil exploration, their predictions of failures, cost overruns and schedule overruns are routinely better than human intuition exploration and oil exploration, their predictions of failures, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods. or alternative "soft" methods.  The Monte Carlo method was coined in the 1940s by John von Neumann, Stanislaw Ulam and Nicholas Metropolis, while they were  The Monte Carlo method was coined in the 1940s by John von Neumann, Stanislaw Ulam and Nicholas Metropolis, while they were working on nuclear weapon projects in the Los Alamos National Laboratory. It was named in homage to Monte Carlo casino, a famous working on nuclear weapon projects in the Los Alamos National Laboratory. It was named in homage to Monte Carlo casino, a famous casino, where Ulam's uncle would often gamble away his money. casino, where Ulam's uncle would often gamble away his money. Introduction and example: Monte Carlo method applied to approximating the value of π Introduction and example: Monte Carlo method applied to approximating the value of π  Monte Carlo methods vary, but tend to follow a particular pattern:  Monte Carlo methods vary, but tend to follow a particular pattern: – Define a domain of possible inputs. – Define a domain of possible inputs. – Generate inputs randomly from a probability distribution over the domain. – Generate inputs randomly from a probability distribution over the domain. – Perform a deterministic computation on the inputs. – Perform a deterministic computation on the inputs. – Aggregate the results. – Aggregate the results.       For example, given that a circle inscribed in a square and the square itself have a ratio of areas that is π/4, the value of π can be For example, given that a circle inscribed in a square and the square itself have a ratio of areas that is π/4, the value of π can be approximated using a Monte Carlo method: approximated using a Monte Carlo method: – Draw a square on the ground, then inscribe a circle within it. – Draw a square on the ground, then inscribe a circle within it. – Uniformly scatter some objects of uniform size (grains of rice or sand) over the square. – Uniformly scatter some objects of uniform size (grains of rice or sand) over the square. – Count the number of objects inside the circle and the total number of objects. – Count the number of objects inside the circle and the total number of objects. – The ratio of the two counts is an estimate of the ratio of the two areas, which is π/4. Multiply the result by 4 to estimate π. – The ratio of the two counts is an estimate of the ratio of the two areas, which is π/4. Multiply the result by 4 to estimate π. In this procedure the domain of inputs is the square that circumscribes our circle. We generate random inputs by scattering grains over In this procedure the domain of inputs is the square that circumscribes our circle. We generate random inputs by scattering grains over the square then perform a computation on each input (test whether it falls within the circle). Finally, we aggregate the results to obtain our the square then perform a computation on each input (test whether it falls within the circle). Finally, we aggregate the results to obtain our final result, the approximation of π. final result, the approximation of π. To get an accurate approximation for π this procedure should have two other common properties of Monte Carlo methods. First, the To get an accurate approximation for π this procedure should have two other common properties of Monte Carlo methods. First, the inputs should truly be random. If grains are purposefully dropped into only the center of the circle, they will not be uniformly distributed, inputs should truly be random. If grains are purposefully dropped into only the center of the circle, they will not be uniformly distributed, and so our approximation will be poor. Second, there should be a large number of inputs. The approximation will generally be poor if only and so our approximation will be poor. Second, there should be a large number of inputs. The approximation will generally be poor if only a few grains are randomly dropped into the whole square. On average, the approximation improves as more grains are dropped. a few grains are randomly dropped into the whole square. On average, the approximation improves as more grains are dropped. © 2013 IBM Corporation 42

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