“Luck is where preparation and opportunity meet.”
This timeless insight from the Stoic philosopher Seneca offers a framework for understanding how we can harness preparation and opportunity to create success. Yet in modern times, this aphorism needs refinement. To fully capture the mechanics of luck, we must account for adaptability, action, and the uncertainty and imperfect information that shape our decisions about the future. Mathematics provides a powerful lens to explore how preparation, opportunity, and time work together to navigate these challenges and generate extraordinary outcomes.
At the heart of this exploration lies the importance of mathematical intuition. While most people may not regularly practice higher-level math or use formal mathematical notation, it is the underlying intuition that truly matters. Grasping concepts like convexity and other financial principles is essential for understanding how to create and leverage your own luck. These insights are more than theoretical—they offer practical tools to navigate uncertainty, make informed decisions, and maximize opportunities despite incomplete information about what lies ahead.
The good news? You don’t need to be a mathematician to unlock this power. With a strong sense of mathematical intuition, you can extract significant value from these ideas without mastering every technical detail. Think of math as the bridge between abstract relationships and real-world applications. This article aims to arm you with that intuitive understanding.
There will be some math notation—just enough to describe the relationships between the real world and the math that represents it. But the focus remains on providing actionable insights and empowering you to see how deliberate choices, sustained effort, and strategic preparation can make luck an intentional and repeatable phenomenon.
This article delves into the interplay of preparation, opportunity, adaptability, and effort. Using an updated mathematical model and real-world examples, we’ll uncover how deliberate actions and strategic choices can transform “luck” into a repeatable formula for success.
Table of Contents
Introduction
The Intuition Behind Luck and Mathematics
Turning Chance into Strategy
The Foundations of Luck
Preparation and Time
Opportunity and Growth Rate
Adaptability and Preparation Growth
Mathematical Model Overview
Key Variables in the Luck Equation
The Interplay of Preparation, Time, and Opportunity
The Code Learner’s Journey
Harnessing the Winds of Luck in Career Growth
The Garbage Picker’s Transformation
From Determinism to Opportunity
Amplifying Luck: Practical Insights
Raising Preparation Growth (r)
Finding High-Growth Opportunities (β)
Leveraging Nonlinear Scaling (α)
Conclusion
Making Luck Intentional
Appendix
Mathematical Model of Luck Over Time
Visualizing Preparation and Opportunities
About the author: Jeff Hulett leads Personal Finance Reimagined, a decision-making and financial education platform. He teaches personal finance at James Madison University and provides personal finance seminars. Check out his book -- Making Choices, Making Money: Your Guide to Making Confident Financial Decisions.
Jeff is a career banker, data scientist, behavioral economist, and choice architect. Jeff has held banking and consulting leadership roles at Wells Fargo, Citibank, KPMG, and IBM.
2. The Foundations of Luck
Preparation and Time (t)
Preparation forms the foundation of luck—it is the learning, practicing, and skill-building that ensure readiness when opportunities present themselves. However, preparation does not yield immediate results; it requires time (t) to compound. Early efforts may feel incremental, with small gains that seem disproportionate to the effort invested. But over time, consistent preparation creates a compounding effect, accelerating readiness and opening doors to greater opportunities.
Consider a student learning to code. In the beginning, they might struggle to write even basic programs, with hours of effort producing modest progress. However, as their knowledge deepens and they tackle increasingly complex projects, each hour of study becomes more impactful. Skills gained early amplify future learning, and the student’s ability to solve real-world problems grows exponentially. Eventually, this preparation leads to high-value opportunities, such as internships, collaborative projects, or even career advancement.
The same dynamic applies to fitness. A person starting a workout regimen might initially find it difficult to see results—improving endurance or building strength requires weeks of consistent effort. Yet, as their fitness level improves, their capacity to train harder and recover faster also increases. Over time, the compounding effects of preparation and practice lead to significant gains, turning early struggles into measurable success.
Whether applied to coding, fitness, or any other endeavor, preparation and time (t) are inseparable allies. With consistent investment, these small, early steps accumulate into a preparation base that makes higher-value opportunities increasingly attainable. This principle forms the backbone of the mathematics of luck: time amplifies preparation, creating the conditions for exponential growth and success.
Opportunity and Growth Rate (β)
Opportunities often seem unpredictable, but the rate at which they arise (β) depends on how strategically you position yourself. While some opportunities are obvious, others require deliberate effort to uncover and leverage. β reflects the speed at which opportunities grow in response to your environment, actions, and adaptability.
Tina Seelig, a Stanford entrepreneurship expert, emphasizes the importance of creating opportunity-rich conditions, what she calls placing yourself in the “winds of luck.” This involves seeking environments where opportunities are abundant and making yourself visible through active participation. For example, if you are a young professional aspiring to enter the tech industry, engaging in hackathons, contributing to open-source projects, or attending industry meetups can significantly raise your β. These activities connect you with industry leaders, peers, and mentors, exposing you to potential collaborations and job opportunities.
High-β opportunities are not always obvious. Resources like the Bureau of Labor Statistics’ Fastest Growing Occupations list offer insights into industries and roles projected to expand rapidly. For instance, fields like renewable energy, healthcare technology, and artificial intelligence currently exhibit high growth rates. Aligning your preparation with these trends can capture emerging opportunities.
By actively identifying and engaging with high-β environments, you increase the likelihood of encountering transformative opportunities. When paired with strategic preparation and adaptability, this approach ensures that you can effectively harness the “winds of luck” and achieve exponential growth in your professional and personal endeavors.
Adaptability and Preparation Growth Rate (r)
Adaptability, as venture capitalist Natalie Fratto emphasizes, is the critical bridge between preparation and opportunity. It directly influences the preparation growth rate (r), determining how effectively time investments translate into readiness. Adaptability is not just about responding to change but proactively engaging in behaviors that amplify growth, such as asking “What if?” questions, actively unlearning outdated methods, and exploring innovative possibilities. These habits ensure that preparation compounds more efficiently, accelerating progress.
For example, consider a marketing professional navigating the rise of artificial intelligence in their field. By asking “What if AI automates most content creation?” they might identify emerging skills, like data analysis or AI tool mastery, to remain competitive. Engaging in active unlearning, they discard reliance on traditional strategies that AI can outperform. Through exploration, they attend workshops, experiment with AI tools, and collaborate with technologists, rapidly increasing their r.
Fratto highlights that adaptability creates a feedback loop: as readiness improves, individuals quickly spot and seize higher-value opportunities. Contrast this with someone resistant to change, whose growth stagnates because they cling to outdated methods.
Ultimately, adaptability ensures that preparation not only compounds but remains relevant, enabling you to pivot, innovate, and thrive as circumstances evolve. By embracing adaptability, you can amplify your r, transforming challenges into competitive advantages and significantly increasing your likelihood of long-term success.
Nonlinear Scaling (α) vs. Opportunity Growth Rate (β)
Effort and preparation often yield results that are far from linear, but this is not the same as opportunity growth over time. The variable α represents nonlinear scaling, showing how substantial preparation disproportionately enhances the value of opportunities. In contrast, the opportunity growth rate (β) captures how quickly external opportunities arise over time, influenced by factors like networking, visibility, and adaptability. These variables complement each other but reflect distinct dynamics in the process of success, avoiding double counting by isolating their effects.
To illustrate, consider a software engineer learning a rare programming language, such as Rust, which is gaining traction in systems programming. The effort required to master Rust (r) is considerable, reflecting the time and energy invested in preparation. However, the nonlinear scaling (α) of their preparation ensures that the value of this skill is outsized. Mastering Rust enables access to high-demand, niche roles with significant salaries and the potential to lead innovative projects. This is α in action—magnifying the returns on preparation.
Simultaneously, the opportunity growth rate (β) determines how quickly these roles materialize. For instance, participating in Rust developer forums or attending niche tech conferences accelerates β, increasing the frequency and visibility of job openings.
By separating α and β, the model captures both the inherent value of preparation (α) and the pace at which opportunities become accessible (β), illustrating how deliberate effort and strategic exposure combine to create exponential success.
3. Mathematical Model Overview
The mathematical model incorporates preparation Prep(t), opportunity O(t), and time (t) to determine the probability of success P(t) at any moment. It highlights how consistent preparation, over time, enhances readiness, leading to compounding opportunities and an increased likelihood of success.
Additional parameters refine the model and provide valuable intuition: r, the preparation growth rate, captures how adaptability and effort accelerate skill development and readiness. β, the opportunity growth rate, reflects how environments and exposure amplify the emergence of high-value opportunities. The nonlinear scaling factor α accounts for how significant preparation disproportionately magnifies the value of opportunities.
The appendix provides the full formula for those seeking a deeper mathematical understanding, but the key takeaway is this: preparation grows with consistent effort, opportunities flourish when aligned with the right environment, and adaptability and time amplify the overall effect.
In the following sections, we’ll explore how these factors interact, using relatable examples to demonstrate how preparation, time, and environment combine to create exponential growth in opportunities and success. This intuitive framework will guide you in strategically creating and harnessing your own luck.
4. The Garbage Picker’s Transformation: From Determinism to Opportunity
Consider the garbage picker working on a university campus, as highlighted by neuroscientist Robert Sapolsky. In his narrative, the garbage picker’s life appears shaped by deterministic factors—genetics and environment—limiting their trajectory and opportunity for change. While this framing emphasizes the constraints of determinism, it does not account for how preparation and time can create new opportunities, even in challenging circumstances. You may check out the full garbage picker's story here.
Using the updated model, we see that consistent preparation (t) has the power to shift trajectories. The garbage picker who shows up reliably, communicates effectively, and takes advantage of free online courses demonstrates how small, consistent investments in preparation Prep(t) can grow exponentially over time. Early efforts, such as learning basic skills or improving productivity, may seem inconsequential, but they set the stage for compounding growth and a higher probability of success P(t).
This growth translates into real opportunities: a supervisor might notice their dedication and offer them access to a management training program or additional responsibilities. These opportunities (β) are amplified by adaptability (r), ensuring the individual is ready to act when they arise. The garbage picker’s story demonstrates how time and preparation can overcome perceived limitations, transforming even modest beginnings into significant career growth.
5. Amplifying Luck: Practical Insights
Raising r: Adaptability and the Role of Environment
The variable r captures how efficiently preparation compounds over time, and adaptability is a key driver for raising this growth rate. As Natalie Fratto emphasizes, behaviors like asking “What if?” questions uncover hidden opportunities, while active unlearning and exploration ensure preparation evolves with changing demands. These traits foster a higher r, enabling preparation to grow more rapidly and effectively.
The environment is another critical factor in shaping r. A stagnant or restrictive environment suppresses r, constraining growth potential. Conversely, moving to an opportunity-rich setting—such as a dynamic workplace or a forward-thinking community—amplifies r, creating fertile ground for preparation to flourish. Tina Seelig’s “winds of luck” metaphor highlights this: the larger your “sail” (preparation and adaptability), the better you can capture the momentum of environmental opportunities.
Consider a mid-career professional stuck in a static role within a declining industry. By pivoting to a fast-growing sector or joining a more innovative organization, they simultaneously increase r and β, the rate at which opportunities emerge. This strategic shift not only accelerates preparation growth but also ensures they are well-positioned to harness the next wave of high-value opportunities.
Practical Application: Nonlinear Scaling and α
Nonlinear scaling (α > 1) ensures that significant preparation delivers outsized results. This variable highlights the value of specialization. For instance, a chef who masters molecular gastronomy—a niche skill—gains access to high-value roles that generalist chefs cannot compete for. The effort required to reach this level of expertise amplifies r, while α ensures the resulting opportunities are disproportionately rewarding.
The lesson? Focused investments in unique, high-demand skills unlock exponential returns, aligning preparation and opportunity with long-term success.
Conclusion: Luck as a Strategic Choice
Luck may seem like a stroke of randomness, but mathematics reveals its systematic foundation. Preparation, opportunity, and time are the core elements that define success. By investing in preparation, positioning yourself in opportunity-rich environments, and embracing adaptability, you can actively shape your trajectory. The garbage picker’s story exemplifies how consistent effort and strategic choices can elevate even the most constrained circumstances, proving that luck is not merely fate but a process we can influence.
As Seneca’s timeless aphorism suggests, “Luck is where preparation and opportunity meet.” By adding adaptability and effort to this equation, we create a powerful framework for transforming uncertainty into opportunity. For readers eager to solidify their understanding, the appendix offers a deeper dive into the mathematical model, helping you build the intuition needed to apply these concepts to your own life. Harness the math of luck, and take the first steps toward creating your own good fortune.
APPENDIX: Luck Decoded: The Equation Behind Success Over Time
The relationship between opportunity as the numerator and preparation as the denominator is modeled. Preparation (Prep) is functionally related to opportunity O(Prep). Small increases in preparation lead to a disproportionately larger increase in opportunities and thus increase the overall probability (outcome). Time (t) is included to capture the duration of preparation and its cumulative effect on opportunities. A convex relationship between time and preparation as an investment is assumed, much the way regular savings will compound for higher returns. This aligns with neuroscientist Eric Hoel’s concept of Causal Emergence, where small-scale actions, such as consistent preparation, build the foundation for higher-scale outcomes, like transformative opportunities. By compounding over time, preparation amplifies its causal significance, bridging the gap between effort and success.
We introduce a time variable (t) to capture the duration of preparation and its cumulative effect on opportunities:
Time-Dependent Preparation (Prep(t)):
Preparation often grows with consistent time investment, which can be modeled exponentially (reflecting compounding effort):
Time-Dependent Opportunities (O(Prep,t):
Opportunities depend on both preparation and a convex time component. This reflects that opportunities disproportionately increase as preparation matures:
Solve the Probability Formula:
Key Features of the Updated Model:
Insights:
Early Effort Matters: While opportunity growth is slow initially, it compounds over time, emphasizing the value of consistent preparation.
Timing is Critical: The convexity of opportunity growth highlights the importance of sustained investments, even when immediate payoffs are not visible.
Higher Returns: The model mirrors the time value of money, where delayed but consistent preparation leads to exponential increases in the probability of success.
A low preparation and high preparation example
The visualizations compare a "low preparation" scenario to a "high preparation" scenario over time. The following key points highlight how making consistent preparatory investments leads to more opportunities and a higher probability of success:
Preparation Over Time:
Low preparation starts at a smaller baseline and grows slower compared to high preparation.
High preparation exhibits exponential growth over time due to the increased initial investment and compounding effects.
Opportunities Over Time:
Opportunities scale non-linearly with preparation. High preparation results in significantly more opportunities, as illustrated by the steeper growth curve.
Early differences in preparation create increasingly larger gaps in opportunity over time.
Probability of Success:
The high preparation scenario achieves a markedly higher probability of success than the low preparation scenario, particularly over extended time horizons.
This is due to the combined impact of higher preparation levels and the convex nature of opportunity growth relative to preparation investments.
This model underscores the importance of sustained and higher initial investments in preparation to maximize long-term opportunities and success.
Resources for the Curious
Hulett, J. (2023). Luck is where preparation and opportunity meet. The Curiosity Vine. Retrieved from https://www.thecuriosityvine.com
Hulett, J. (2023). Challenging Our Beliefs: Expressing our free will and how to be Bayesian in our day-to-day life. The Curiosity Vine. Retrieved from https://www.thecuriosityvine.com
Dweck, C. S. (2006). Mindset: The new psychology of success. Ballantine Books.
Seelig, T. (2017). Creativity rules: Get ideas out of your head and into the world. HarperOne.
Fratto, N. (2019). 3 ways to measure your adaptability—and how to improve it [Video]. TED. Retrieved from https://www.ted.com
Sapolsky, R. M. (2017). Behave: The biology of humans at our best and worst. Penguin Press.
Hoel, E. (2023). The world behind the world: Consciousness, free will, and the limits of science. Avid Reader Press / Simon & Schuster.
Attia, P. (2023). Outlive: The science and art of longevity. Harmony Books.
Kahneman, D. (2011). Thinking, fast and slow. Farrar, Straus and Giroux.
McGilchrist, I. (2009). The master and his emissary: The divided brain and the making of the Western world. Yale University Press.
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