Betting on Tomorrow: How Probabilities Help You Make Smarter Decisions
- Aran Kaila
- Aug 1
- 12 min read
Every day, we navigate a world brimming with uncertainty. From deciding whether to carry an umbrella to choosing an investment option, life presents a continuous stream of decisions, each with potential outcomes we instinctively weigh. This inherent human tendency to consider possibilities, even if informally, suggests we already engage in a rudimentary form of probabilistic thinking. At "Built on Numbers," our aim isn't to introduce an entirely foreign concept, but rather to formalise and enhance this existing cognitive process. By providing explicit tools and frameworks, we can refine your ability to make more robust decisions.
This article will introduce two powerful numerical tools – probability and expected value – that can transform how you approach decisions. We'll delve into how understanding the likelihood of various outcomes can inform better everyday choices, helping you "bet" more wisely on your future. We believe that a little numerical literacy can go a long way in empowering you to navigate life's complexities with greater confidence, turning uncertainty into informed action.
The Fundamentals: Understanding Probability and Expected Value
Probability Demystified
At its heart, probability is simply a numerical measure that quantifies how likely an event is to occur. It’s always expressed as a number between 0 and 1 (or 0% and 100%), where 0 means it’s impossible, and 1 means it’s a sure thing. This range gives us a universal scale for assessing likelihood.
The easiest way to calculate the probability of an event is to divide the number of 'favourable outcomes' (the specific outcomes you're interested in) by the 'total number of possible outcomes'. For example, if you roll a fair six-sided die, the probability of rolling an even number (2, 4, or 6) is 3 (favourable outcomes) divided by 6 (total outcomes), giving you 3/6 or 1/2. Similarly, the probability of rolling a prime number (2, 3, or 5) is also 3/6 or 1/2.
A crucial concept for assessing risk is the complement rule: the probability of an event not happening is simply 1 minus the probability of it happening. For instance, if the Met Office forecasts a 0.3 (or 30%) probability of rain, the probability of it not raining is 1 - 0.3 = 0.7 (or 70%). Understanding the probability of an event not happening is just as vital as knowing the probability of it happening, especially when you're planning and trying to mitigate risks. This simple yet profound relationship forms the bedrock of practical risk assessment, enabling a more nuanced understanding of binary choices.
The 'sample space' refers to the set of all possible outcomes of an experiment. For example, a single coin toss has a sample space of {Heads, Tails}. If you toss two coins simultaneously, the sample space expands to four possible outcomes: (Heads, Heads), (Heads, Tails), (Tails, Heads), and (Tails, Tails).
Expected Value: Beyond Simple Chance
Expected Value (EV) isn't a prediction of what will happen in a single instance. Instead, it represents the long-term average outcome if a random process were to be repeated numerous times. Think of it as a weighted average of all possible values, where each value is weighted by its probability of occurrence.
The core formula for EV is calculated by multiplying each possible outcome by its respective probability and then summing these products. The formula is: EV = Σ [Outcome (Xᵢ) * Probability (P(Xᵢ))].
The intuition behind EV is powerful: if you repeatedly engage in a scenario with a positive EV, you can expect to profit over the long term. Conversely, a negative EV implies an expected loss over many repetitions. This concept is crucial for making informed decisions, especially when multiple outcomes are possible and decisions are repeated. It’s important to note that EV provides guidance for a long-term strategy and expected average outcomes over many trials, rather than a prediction for any single occurrence. This distinction helps manage expectations and prevents misapplication of the concept, particularly for one-off, high-stakes decisions.
Let’s consider a simple game where you pay £1 to play. You roll a fair six-sided die. If a 6 is rolled, you win £4 (your £1 stake back plus £3 profit). If any other number (1, 2, 3, 4, or 5) is rolled, you lose your £1 stake. The expected value calculation is shown below:
Outcome (Net Gain/Loss in £) | Probability P(X) | X * P(X) |
Win (£3 profit) | 1/6 | £0.50 |
Lose (£-1 stake) | 5/6 | £-0.83 |
Total Expected Value (Sum) | £-0.33 |
This table clearly illustrates that for every £1 wagered in this game, you can expect to lose, on average, £0.33 over many plays. This makes it a negative EV game, indicating it’s not a financially sound decision to play it repeatedly.
Navigating Everyday Uncertainty: Calculated Risks in Daily Life
Below, we explore applications of these fundamental concepts of probability to common and relatable scenarios.
Weathering the Storm
The UK Met Office, a primary source for weather information, extensively uses probabilities in its forecasts. Instead of a simple "it will rain", they provide percentages (e.g., "20% chance of rain") or ranges. This indicates that, based on historical data and complex models, similar atmospheric conditions have historically led to rain a certain percentage of the time. They achieve this through 'ensemble forecasts', running multiple models and estimating an event's probability by counting how many models predict it.
Beyond simple percentages, the Met Office offers detailed probability maps using tercile (33.3% baseline probability) and quintile (20% baseline probability) categories. These indicate the enhanced or depressed likelihood of "out-of-the-ordinary" temperature or precipitation, with colour shading (yellow/orange for higher probability, blue for lower) providing a visual guide.
Understanding these probabilistic forecasts allows for informed decision-making. If there's a 70% chance of rain, you're far more likely to pack an umbrella or adjust outdoor plans than if it's a 10% chance. This approach isn't about absolute certainty, but about intelligently managing the risk of being unprepared. This highlights how probabilistic information isn't merely descriptive; it's actionable intelligence. Knowing the likelihood of rain or traffic delays empowers you to take pre-emptive actions, shifting decision-making from reactive responses to proactive risk management.
Traffic
Traffic prediction in the UK, particularly near junctions, leverages probability and mathematical modelling. These models analyse total traffic flow and average speed data, often from sources like Highways England, to predict congestion patterns. Such predictions can even be used to prioritise emergency vehicles. More sophisticated approaches, including AI and machine learning, are employed to enhance the prediction and analysis of UK road traffic, such as accident severity. While focusing on accidents, the underlying principle of using data and probability to predict traffic conditions (e.g., delays, flow, congestion hotspots) remains consistent.
Knowing the probability of heavy traffic on a specific route (e.g., the M6, a notorious UK motorway) or at a particular time of day can significantly influence your travel choices. This might mean adjusting your departure time, choosing an alternative route, or opting for public transport, all based on a calculated likelihood of a smoother journey.
Event Cancellations
Events, such as a local parkrun, can be cancelled due to various factors, most commonly adverse weather conditions (like flooding, ice, or high winds making the course unsafe) or insufficient volunteers. The probability of such disruptions can be assessed based on historical data for a specific location or real-time conditions.
The likelihood of an event being cancelled can often be conditional on another event occurring. For example, the probability of a sporting event being called off due to a waterlogged pitch is conditional on significant heavy rainfall occurring beforehand. This demonstrates that probabilities in real-world scenarios are rarely isolated; the likelihood of one event can be dependent on the occurrence of another.
As per the conditional probability rule, if Event A (heavy rain) has a 0.6 chance and Event B (cancellation due to waterlogged pitch) has a 0.75 chance given that heavy rain, then the overall probability of both events occurring is 0.6 multiplied by 0.75, which equals 0.45. This recognition of dependent probabilities is crucial for accurate real-world risk assessment.
If a specific parkrun course is historically prone to flooding and a forecast indicates a high probability of heavy rain, the overall probability of cancellation increases. This knowledge allows for alternative plans, such as checking official announcements before leaving home.
Smart Decisions: Money, Health, and Your Future
Now, let's apply both probability and expected value to more significant life decisions, which help to really demonstrate their power in navigating complex choices across personal finance, health, and investments.
Purchasing Power: When the Odds are in Your Favour (or Not)
Credit Cards: Many UK credit cards entice users with cashback (e.g., 0.25% on eligible purchases) or loyalty points. To truly assess the benefit, the expected value of these rewards needs to be calculated. For instance, if you spend £10,000 annually on a card offering 0.25% cashback, the expected annual reward is £25. If the card has no annual fee and the balance is consistently paid off in full to avoid interest, this represents a clear positive expected value.
Betting: In the realm of sports betting, Expected Value (EV) is a critical metric, calculated as: (Winning Amount per Bet x Winning Probability) – (Lost Amount per Bet x Losing Probability).
A positive EV bet indicates that, over the long run, you can expect to profit, even if individual bets do not always win. Positive EV opportunities arise when the estimated "true probability" of an event occurring is higher than the implied probability offered by the bookmaker's odds.
For example, if Coventry City has decimal odds of 2.78 to win a football match (implying a 36.23% probability), but your analysis suggests their true probability of winning is 40%, then a £50 bet on Coventry would yield a positive EV. The example for a £50 stake on Coventry calculates a negative EV of -£0.60, indicating it's not a profitable long-term bet.
Successful EV betting demands discipline. It requires consistently placing bets with a positive expected value, understanding that short-term losses are an inherent part of the process, but long-term profitability is the ultimate goal.
Job Offers: A job offer extends far beyond the headline salary. It encompasses the total expected value of the entire remuneration package. This includes mandatory UK benefits such as employer National Insurance contributions (typically 13.8% of earnings), workplace pension contributions (usually at least 3% from the employer), statutory sick pay (£109.40 per week for up to 28 weeks), and statutory annual leave (28 days paid).
While more challenging to quantify precisely, non-monetary benefits like private healthcare, additional annual leave, flexible working arrangements, or professional development opportunities significantly contribute to the overall expected value of an offer.
You can assign a subjective probability and value to these based on your personal priorities. By calculating the comprehensive expected value of different job offers, a more informed decision can be made that goes beyond just the salary figure. For instance, a slightly lower salary coupled with significantly better pension contributions, enhanced health benefits, and generous leave might present a higher overall expected value for long-term well-being.
While salary surveys and recruitment companies provide valuable market benchmarks, the true expected value of a job offer necessitates a comprehensive assessment of all tangible and intangible benefits, weighted by their likelihood of being utilised and their personal utility.
Insurance: Insurance companies are fundamentally built on the principles of probability. They utilise sophisticated calculations, drawing on extensive historical data and risk assessments, to determine the probability of an insured event (e.g., a car accident, house fire, or medical emergency) occurring. Based on these probabilities, they set premiums that aim to cover potential payouts, operational costs, and generate profit. The premium you pay is essentially the cost of transferring the financial risk of that event to the insurer.
From an individual's perspective, the pure mathematical expected value of an insurance policy is typically negative (the premium paid generally exceeds the expected payout) because the insurer must cover its expenses and make a profit. However, the true value for the insured lies in mitigating the risk of catastrophic financial loss.
For example, a "Whole of Life" insurance policy's value can be demonstrated by comparing the total premiums paid against the potential payout, also factoring in an assumed interest rate if that money were simply saved (i.e. opportunity cost).
This highlights a crucial distinction in decision-making. While the mathematical expected value of an insurance policy for the individual is often negative, its utility or peace of mind value is exceptionally high. This demonstrates that not all "smarter decisions" are solely driven by maximising a positive expected monetary value; often, they are driven by risk aversion – the desire to avoid low-probability, high-impact negative outcomes.
This pattern indicates that while Expected Value provides a robust quantitative framework, human decision-making inherently incorporates qualitative factors, personal preferences, and varying degrees of risk aversion.
Health Choices: Informed Decisions for a Healthier You
Lifestyle Changes: Public health data from the UK consistently demonstrates that lifestyle changes, such as adopting a healthier diet, losing weight, increasing physical activity, or quitting smoking, significantly increase the probability of improved health outcomes. For instance, smoking is identified as the largest preventable cause of ill health and premature mortality in England.
While precise individual probabilities are challenging, population-level statistics provide strong probabilistic evidence. For example, people living in areas with lower healthy life expectancy are 1.7 times more likely to smoke. This implies that reducing smoking significantly increases an individual's probability of a longer, healthier life.
This highlights that even though health statistics are often aggregated at a population level, they provide compelling probabilistic evidence that can inform and motivate individual lifestyle choices. The "average" benefit observed in large populations translates into an increased personal probability of achieving better health outcomes.
Medical Treatments: Medical professionals and patient information leaflets (PILs) are key sources for understanding the probabilities associated with medical treatments. They provide data on treatment success rates and the likelihood of various side effects. For example, a PIL will categorise side effects based on their probability of occurrence.
Deciding on a medical treatment involves a careful weighing of the probability of the treatment's success against the probabilities and potential severity of its side effects. Chemotherapy, for instance, offers a probability of destroying a tumour but also carries a high probability of adverse effects like fatigue, nausea, and hair loss. Advanced clinical trials often employ Bayesian statistics to calculate the probability of success, incorporating external data sources (such as previous trials or real-world data) to refine these probability calculations.
This allows for a more nuanced and dynamic understanding of treatment efficacy. While medical research provides robust population-level probabilities, the individual patient's decision-making process involves interpreting how these general probabilities apply to their unique health profile and circumstances, often in consultation with healthcare professionals. This underscores the inherent challenge of applying population-based inferences to individual patients, even when the underlying epidemiological data is accurate.
Investing Basics: Balancing Risk and Return
Probability of Return vs. Risk: All investments inherently carry an element of risk – the possibility that an investment may lose value. As a general rule, higher potential returns are associated with higher levels of risk, meaning a greater probability of losing value. For example, the FTSE 100, a benchmark for UK economic performance, has generated an average annualised return of 6.3% (including dividends) over the past 20 years. However, this has not been without significant volatility, including crashes of 20% during the dot-com bubble and 25% during the Global Financial Crisis.
It is crucial to consider "real returns" when evaluating investments. This is calculated by subtracting the rate of inflation from the nominal return, as inflation erodes the purchasing power of money over time.
While not always explicitly labelled "Expected Value," calculating Return on Investment (ROI) in finance involves assessing potential gains against costs. For property investments, a simple ROI calculation might be Annual Rent divided by Purchase Price, multiplied by 100 to get a percentage. More broadly, ROI can be calculated as (Net Return on Investment / Cost of Investment) * 100%. These calculations help determine the expected profitability.
The Power of Diversification: Diversification is a core investment strategy, encapsulated by the adage, "Don't put all your eggs in one basket." By investing across a variety of different securities or asset classes (e.g., equities, bonds, real estate), the aim is to reduce overall portfolio volatility and mitigate the probability of significant losses.
Different asset classes tend to behave differently under various economic conditions. While higher-risk assets like equities (e.g. individual stocks) can experience substantial falls in value (e.g., 50% or more), a well-diversified portfolio aims for more consistent returns by balancing these with lower-risk assets that may act as a buffer during downturns.
Academic research strongly supports the idea that asset allocation – the strategic decision of how to spread investments across different asset types – is the single most important factor determining a diversified portfolio's long-term performance, accounting for as much as 94% of results.
This is a probabilistic strategy designed to increase the likelihood of achieving long-term financial objectives while effectively managing potential losses during periods of market volatility. The accuracy and reliability of any probabilistic assessment are directly contingent upon the quality, relevance, and completeness of the underlying data. If the input data is biased, outdated, or insufficient, the calculated probabilities and expected values will be flawed, leading to potentially poor decisions.
In investing, the application of probability extends beyond merely predicting the movement of a single stock. It becomes a fundamental driver for designing an overarching portfolio strategy that optimises the probability of achieving long-term financial goals while simultaneously managing the probability of significant downside risk. This represents a higher-order application of probabilistic thinking in a complex domain.
Final Thoughts: Your Toolkit for a Smarter Future
Life is undeniably a series of calculated risks. As we've explored, probability and expected value are not merely abstract mathematical concepts confined to textbooks. They are powerful, practical tools that empower you to make more informed and intelligent decisions across every facet of life – from daily planning and purchasing choices to critical health considerations and long-term investments.
By consciously understanding the likelihood of various outcomes and the long-term average value of different choices, you move beyond mere guesswork and reactive responses. You gain the invaluable ability to assess opportunities and risks with greater clarity and confidence, helping you to "bet" more wisely on your future.
At "Built on Numbers," we're dedicated to providing the insights and tools necessary to make those calculations a little smarter. Embracing the power of numbers can transform life's inherent uncertainties into opportunities for informed, strategic action.
Start small, apply these concepts to everyday decisions, and gradually build confidence in navigating life's complex, probabilistic landscape. It's how you build a smarter future, one calculated step at a time.
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