Tag: probability

Risk: an introduction

Risk of an event = Probability of the event happening × the consequensces of the event happening.1

To understand probability better, please read this and this.

This is the most basic definition of Risk. Risk = Probability, or how likely an event is to occur × Consequence, or impact. Because it is multiplicative, a high-probability event with low consequence (losing a pen) is low risk, and a low-probability event with catastrophic consequence (say, a nuclear exchange) can be high risk. The danger zone is where meaningful probability meets serious consequence.

History
For most of history, people spoke about fate, luck, or divine will, not “risk” in a calculable sense. Hazards (storms, plagues, crop failures) were seen as acts of gods or nature. There was no notion of systematically measuring uncertainty.

In the 17th Century, A French nobleman, Chevalier de Méré, asked Blaise Pascal why some gambling bets worked better than others. Pascal’s correspondence with Pierre de Fermat (1654) is widely seen as the birth of modern probability theory.23 They developed early ideas of expected value – essentially, the mathematical ancestor of “probability × impact”.4

In the 18th Century, Daniel Bernoulli introduced the idea of utility in 1738:5 the insight that losing or gaining the same amount (£100) does not feel equally important to rich and poor people. This work planted the seeds for understanding why humans are risk‑averse and set the stage for later behavioural theories.

As trade, shipping and life insurance developed in the 18th–19th centuries, people started using probability tables to price the risk of death, shipwrecks and fire.6 This was the first large‑scale, institutional attempt to put numbers on everyday risks and pool them.6 Risk pooling is when lots of people chip in a little money into a shared pot (the “pool”) so that when one person has a big, unexpected cost (like a car accident or sickness), the money from the whole group covers it, making big losses manageable for individuals and premiums more stable for everyone.7 After industrialisation, wars and technological disasters, “risk” broadened from individual hazards (a ship sinking) to complex systems (nuclear power, financial markets, supply chains). The language of “risk management” emerged after the Second World War and matured through the later 20th century, culminating in general standards such as ISO 31000.89

Expected Value910
The mathematical heart of risk is Expected Value (EV). This is simply the average outcome if you repeated an action infinitely.

If a bet offers a 50% chance to win £100 and a 50% chance to lose nothing, the Expected Value is £50 ($0.50 \times 100 + 0.50 \times 0$). Rationally, you should pay anything up to £49.99 to take that bet.

But real life isn’t a casino with infinite replays. Humans often get only one shot. If an individual takes a risk with a positive expected value—like cycling to work to save money and improve health—but gets hit by a bus on day one, the “average” outcome is irrelevant. This is why variance matters as much as the average. A risk might look good on paper (high expected value) but have a “ruin condition” (a consequence you can’t recover from) that makes the math irrelevant.

Normal Distribution
If you measured the height of every single individual on the planet, or even a representative sample of them, the shape of that graph (often called “curve” in academic language) would be similar to this image:

Normal Distribution.11

This is the Normal Distribution (or Bell Curve), and it is the most important shape in risk management.12 It describes how randomness usually behaves. The very top of the hill represents the Mean (the average). This is what you “expect” to happen; in our stadium example, this is the average height (say, 5’9″). The vast majority of people will be average height, so their heights will be recorded as being clustered right around the middle.

If the Mean tells you where the peak is, Variance tells you how wide the hill is. It is a statistical measure showing how spread out a set of data points are from their average.13

  • Low Variance: Imagine a hill that looks like a needle. This means data points are tightly clustered. If you measured the height of 10,000 professional jockeys, the variance would be low—almost everyone is close to the average.14
  • High Variance: Imagine a hill that looks like a flattened pancake. This means data is widely spread out. If you measured the height of a random crowd containing jockeys and basketball players, the hill would be very wide.15

In risk management, mean tells you what usually happens; variance measures unpredictability and the potential for outcomes to be very different from the average, which is the essence of uncertainty.1617 A high variance means numbers are widely scattered, increasing the chance of both extreme positive and, crucially, extreme negative outcomes (losses).18 Low variance indicates they are clustered closely around the mean: it quantifies the dispersion or variability within a dataset.18 In the height data set, while most people would be average height, some people would be very short and others very tall as well. It’s just that the number of people who are not close to the average would fall off the farther away we get from the mean, or the middle of the bell curve.

Standard Deviation1819

Normal Distribution divided into standard deviations distances from the mean.20

If Variance tells you the hill is “wide,” Standard Deviation (Sigma, or σ) tells you exactly how wide in real units. It is simply the square root of variance.

Think of Standard Deviation as the ruler for the Bell Curve.

  • 1 Standard Deviation: In a normal distribution, about 68% of all outcomes happen within one standard deviation of the mean. If the average height is 5’9″ and the standard deviation is 3 inches, 68% of men are between 5’6″ and 6’0″.
  • 2 Standard Deviations: Go out a bit further, and you capture 95% of all outcomes.
  • 3 Standard Deviations: Go out three steps, and you capture 99.7% of everything.

In risk, when someone talks about a “Six Sigma” event (six standard deviations away from the average), they are talking about something so rare that it should theoretically almost never happen. And yet, in financial markets and complex systems, these “impossible” events happen surprisingly often.

Confidence2122
If a bank says, “We are 95% confident we won’t lose more than £1 million tomorrow,” they are essentially saying: “If tomorrow is a normal day (one of the 95%), we are safe. But if tomorrow is one of those rare, 1-in-20 bad days, all bets are off.”

In statistics, confidence is often explained using confidence intervals: at a 95% confidence level, the method used to build the interval would capture the true value about 95 times out of 100 repeated samples. That does not mean the true value has a 95% probability of being inside this specific interval; it means the procedure has 95% long-run reliability. This means, confidence intervals speak about frequency: how often do the unexpected or unwanted events happen. At 95%, they happen on any 5 days out of 100. at 99%, they happen once every 100 days.

For risk management, think of confidence levels as a dial for paranoia:

  • 95% Confidence: You are planning for the normal bad days. You accept that on 1 day out of every 20 (roughly once a month), you will breach your limit.
  • 99% Confidence: You are planning for the severe days. You only accept breaching your limit on 1 day out of 100 (roughly 2–3 times a year).
  • 99.9% Confidence: You are planning for near-disaster. You only accept a breach once every 1,000 days (roughly once every 4 years).

The Micromort
In the 1970s, Stanford professor Ronald Howard needed a way to compare diverse risks like skydiving, smoking, and driving. He invented the Micromort—a unit representing a one-in-a-million chance of death.23

This equalises different activities. Instead of vague fears (“is it safe to fly?”), we can use units:

  • 1 Micromort is roughly the risk of driving 250 miles (400 km).24
  • 1 Micromort is also the risk of flying 6,000 miles (9,600 km).24
  • Scuba diving costs about 5 micromorts per dive.25
  • Skydiving costs about 8–10 micromorts per jump.24
  • Just being alive (all-cause mortality for a young person) costs roughly 1 micromort per day.26

In conclusion, risk is the price of life.

Sources

  1. ISO 31000 Risk Management Process – Practical Risk Training
  2. July 1654: Pascal’s Letters to Fermat on the “Problem of Points” – APS News
  3. How a Letter Between Two Mathematicians in 1654 Changed the Way We View the Future – KPBS
  4. Pascal and Fermat (1654) – Ebrary
  5. Daniel Bernoulli (1738): Evolution and Economics Under Risk – UBC Zoology (PDF)
  6. The History of Insurance: From Ancient Risk to Modern Protection – Briggs Agency
  7. Risk Pooling: How Health Insurance Works – American Academy of Actuaries
  8. The Evolution of Risk Management: Lessons from History – Risk Management Strategies
  9. Expected Value Calculator – Omnicalculator
  10. Expected Value in Statistics: Definition and Calculation – Statistics How To
  11. Introduction to Gaussian Distribution – All About Circuits
  12. Empirical Rule (68-95-99.7) Explained – Built In
  13. Calculate Standard Deviation & Variance – SurveyKing
  14. What is considered a high or low variance? – Reddit r/mathematics
  15. Variance in Statistics – GeeksforGeeks
  16. Risk-Managing the Uncertainty in VaR Model Parameters – Research Affiliates (PDF)
  17. The Risks of Uncertainty – ACCA Global
  18. Variance – GeeksforGeeks
  19. Empirical Rule: Definition & Formula – Statistics by Jim
  20. Normal Distribution Diagram – TikZ.net
  21. Definition: Confidence Level – Statista
  22. The Role of Confidence Levels in Statistical Analysis – Statsig
  23. There’s a Small Chance This Article May Kill You (Micromorts) – Portable Press
  24. Quantifying Risk – GS Trust Co
  25. Understanding DAN’s Accident Data – Alert Diver Magazine
  26. Microlives: A Lesson in Risk Taking – BBC Future

A probability analysis of India’s men’s cricket coin toss losses – II UPDATED 25/10/2025

NB: This post is now updated to include the 18th consecutive toss loss.

It’s come to my attention that we have lost the last 17 18 coin tosses in One Day International matches for men’s cricket,1 so here’s a continuation of our unfortunate probabilities.

Here’s a more detailed explanation of probability and our toss-losing powers. This post is a continuation of the linked post, so please read that first. However for the lazy buggers who won’t:

  1. Every coin toss is considered an independent event- the outcome of one fair coin toss will not have any impact on the outcomes of any other fair coin tosses.
  2. The probability of two independent events happening at the same time is the product or multiplication of the probabilities of the two events in question. This is called “joint probability”, so If event A has probability P(A) and event B has probability P(B), and their outcomes do not affect each other, the probability that both occur is P(A) × P(B).
#DateOpponentVenueCaptainToss Result
1Nov 19, 2023AustraliaAhmedabadRohit SharmaLost
2Dec 17, 2023South AfricaCenturionKL RahulLost
3Dec 19, 2023South AfricaGqeberhaKL RahulLost
4Dec 21, 2023South AfricaPaarlKL RahulLost
5Feb 6, 2024EnglandHyderabadRohit SharmaLost
6Feb 9, 2024EnglandVisakhapatnamRohit SharmaLost
7Feb 12, 2024EnglandRajkotRohit SharmaLost
8Aug 10, 2024Sri LankaColomboRohit SharmaLost
9Aug 12, 2024Sri LankaPallekeleRohit SharmaLost
10Aug 15, 2024Sri LankaDambullaRohit SharmaLost
11Feb 20, 2025BangladeshDubaiRohit SharmaLost
12Feb 23, 2025PakistanDubaiRohit SharmaLost
13Mar 2, 2025New ZealandDubaiRohit SharmaLost
14Mar 4, 2025AustraliaDubaiRohit SharmaLost
15Mar 9, 2025New ZealandDubaiRohit SharmaLost
16Oct 19, 2025AustraliaPerthShubman GillLost
17Oct 23, 2025AustraliaAdelaideShubman GillLost
18Oct 25, 2025AustraliaSidneyShubman GillLost
India’s 17 18 consecutive ODI coin toss losses in men’s international cricket

You’ll notice that once again the tosses have been lost across tournaments, three different captains, and multiple venues (home and away), and the calling captains choosing heads or tails at random and India still losing every time.

Now, at first I thought that the all format streak of losing 16 consecutive tosses and this ODI streak of losing 17 consecutive tosses were just one series of unfortunate events, but now I want to understand what the probability is of these being considered separate streaks and both “events” still occurring.

So here are the two overlapping streaks:

  1. The ODI-specific streak (Nov 2023–Oct 2025): 17 18 consecutive ODI toss losses.
    Probability = (1/2)^17 = 1/131,072 ≈ 0.00076% (1/2)18 = 1/262,144 ≈ 0.000381%; and​
  2. The all-format streak (Jan–Oct 2025): 16 consecutive toss losses across formats. Probability = (1/2)16 = 1/65,536 ≈ 0.0015%.​

And the probability that these two have coexisted is just the multiplication of the two independent streaks, which is P = (1/131072) × (1/262,144) = 1/8589934592, or about 1/8,600,000,000, which is one in 8.6 billion 1/17179869184, or about 1/17,000,000,000, which is one in 17 billion.

As of mid-2025, the world population was estimated to be around 8.2 billion.2 So if in the middle of this year, if every single person had tossed a fair coin TWICE, there is a possibility that these two streaks would still not have overlapped. It’s an astronomical rarity, so of course we’re on the wrong side of it, *depressed emoji*.

In probability theory, there is a concept of waiting time. Waiting time in streak probability asks how long before you see the streak in question happen? So here it will ask, “How many tosses, on average, until you first see a streak of n consecutive heads (or losses, or wins)?” For a fair coin, the expected number of tosses (waiting time) to see an uninterrupted streak of length n is approximately: En = 2(n+1) – 2.3

In the formula, “n” is the length of the streak.

For a streak of 6 coin toss losses, we will have to wait for

E6 = 2(6+1) – 2

E6 = 27 – 2

E6 = 2 × 2 × 2 × 2 × 2 × 2 × 2 – 2

E6 = 128 – 2 = 126 coin tosses.

  • So, for our first streak of 16 consecutive coin toss losses, the world waited with bated breath for 217 – 2 = 131,070 fair tosses;
  • For the ODI 17 18 coin toss loss streak, we waited for 218 − 2 = 262,142 219 -2 = 524,286 fair tosses; and
  • For both to happen together, we waited 131,070 × 262,142 524,286 fair tosses, or 68,718,166,020, or more than 34 68.7 billion fair coin tosses- A NUMBER SO WILD (okay, calm down, calm down) even cricket fans don’t expect it.

What the hell, my guys?

NB: I just realised that the most widely accepted scientific estimate for the age of the known universe is about 13.8 billion years,4 so the chances of these two streaks happening at all, let alone together, actually involves numbers several times greater than the entire age of the universe in years. Personal suggestion to Shubman Gill- havan karwale bhai.

Sources

  1. A 1 in 130,000 chance: India extend world record ODI toss losing streak to 17 matches
  2. World Population Day: trends and demographic changes
  3. How many coin flips on average does it take to get n consecutive heads?
  4. How old is the universe?

A probability analysis of India’s men’s cricket coin toss losses

India has now lost 16 consecutive coin tosses across all formats, with the streak extending from January 31, 2025, to October 2, 2025 (the West Indies – India Test match in Ahmedabad that concluded today). Here’s the baffling list by chronology:

Coin Toss Loss No.DateMatchVenueCaptain
1Jan 31, 20254th T20I vs EnglandPuneSuryakumar Yadav
2Feb 02, 20255th T20I vs EnglandMumbai (Wankhede)Suryakumar Yadav
3Feb 06, 20251st ODI vs EnglandNagpurRohit Sharma
4Feb 09, 20252nd ODI vs EnglandCuttackRohit Sharma
5Feb 12, 20253rd ODI vs EnglandAhmedabadRohit Sharma
6Feb 20, 2025ODI vs BangladeshDubai (Champions Trophy)Rohit Sharma
7Feb 23, 2025ODI vs PakistanDubai (Champions Trophy)Rohit Sharma
8Mar 02, 2025ODI vs New ZealandDubai (Champions Trophy)Rohit Sharma
9Mar 04, 2025ODI vs AustraliaDubai (Champions Trophy Semi-final)Rohit Sharma
10Mar 09, 2025ODI vs New ZealandDubai (Champions Trophy Final)Rohit Sharma
11Jun 20, 20251st Test vs EnglandLeedsShubman Gill
12Jul 02, 20252nd Test vs EnglandBirminghamShubman Gill
13Jul 10, 20253rd Test vs EnglandLord’sShubman Gill
14Jul 23, 20254th Test vs EnglandManchesterShubman Gill
15Jul 31, 20255th Test vs EnglandThe OvalShubman Gill
16Oct 02, 20251st Test vs West IndiesAhmedabadShubman Gill
Indian Men’s toss losing streak

In mathematics, probability measures how likely an event is to occur, and it’s always expressed as a number between 0 (will never happen) and 1 (will definitely happen every time). For a standard fair coin toss, the probability of either heads or tails is exactly 0.5 (or 50%). This is because there are two possible and equally likely outcomes: the coin will either flip to heads or tails (not counting the vanishingly small number of times it may fall on its edge, in which case the toss will be repeated until a result is achieved anyway).

Every toss is also independent, which means that the result of one toss will have no impact on the result of any other toss. When events are independent, the probability of several events occurring in succession is the product (multiplication) of their individual probabilities. So, the probability of losing (or winning) two fair tosses in a row is: Probability of 2 losses = 0.5 × 0.5 = 0.25.

The probability of losing (or winning) 3 fair tosses in a row is therefore = 0.5 × 0.5 × 0.5, which is 0.125.

We’ve lost 16 consecutive tosses across formats, geographies, and captains. The probability of winning or losing a fair coin toss is 0.5 or 1/2. Which means the probability of losing 16 consecutive fair coin tosses is… (0.5)16, which equals 1 in 65,536, or ≈0.0000152588%.

Now, it really must be noted that a cricket coin toss is quite different from a simple game of coin toss between two people (though the mathematics remains exactly the same). The Indian skippers were not always the ones tossing the coin, neither were they always the ones calling heads or tails. In cricket, the standard procedure is that the host captain tosses while the visiting captain calls. However, at neutral venues where neither captain is the host, the procedure varies: a neutral party such as a match official or invited dignitary may toss the coin, or one of the captains may be chosen to toss, or tournament regulations may specify the exact protocol. This means India’s losing streak has transcended not just different formats, captains and venues, but also different toss procedures, making it an even weirder demonstration of statistical randomness.

I decided to investigate the mathematics of this absurdity.

0.0000152588%
How rare is a 0.0000152588% chance of any event happening? Well, more people are struck by lightning annually,1 but fewer people are likely to die by meteorite strike2.

Similar things have happened in cricket before- The Netherlands have previously lost 11 consecutive tosses, and and several teams have lost 9 in a row.3 Rohit Sharma himself has lost 12 consecutively (equalling Brian Lara).3

Independence and the Gambler’s Fallacy
The Gambler’s Fallacy is the (mistaken) belief that because India “lost so many times in a row,” they’re “due” for a win, but since each coin toss is independent and past outcomes have absolutely no impact on the next. Each toss remains a 50-50 chance, regardless of what’s happened before.

The Law of Large Numbers and the Nature of Streaks
The Law of Large Numbers states that if an independent act is performed enough times, the outcomes of this independent event (the coin toss in our case) will eventually (that is, in the long term, given a large number of coin tosses) match the predicted probable outcome of that event (that is, 50% of the times the coin will flip heads, and 50% of the times it will flip tails), but this will of course include every coin toss ever, and not restrict itself to India’s male cricket captains.

This simply means that though the average outcome will even out to about 50% wins and losses, streaks such as 16 losses in a row are still possible, just extremely unlikely. Given enough cricket matches played, even “impossible” events are destined to surface from time to time. Cricket tosses represent a relatively small sample size in the grand scheme of probability. Even if we consider all international cricket matches ever played, this would still represent a small enough sample size where unusual streaks can and will occur (to understand this, compare every cricket toss to every coin toss that has ever happened in history).

Information Theory
In Information Theory, the rarer an event is considered, the more surprising it is found to be. This means losing one toss is not surprising since there is a 50% chance of losing any one random fair toss. However, losing 16 tosses in a row must be considered very surprising because it involves the following outcomes:

Lose the first toss (50% probability), then lose the second toss (50% probability), then lose the third toss (50% probability), then lose the fourth toss (50% probability), then lose the fifth toss (50% probability)… then finally lose the 16th toss, also with a fifty percent probability that you could win it or lose it.

Which means that if nothing else, at least my bewilderment at the streak is justified.

Sources:

  1. What are the chances of being struck by lightning?
  2. What are the Odds a Meteorite Could Kill You?
  3. Most consecutive toss losses in ODIs, full list: India extend all-time world record