The Perception of Probability

  • Updated
  • By Phil Blackwell

An interview with Professor David Hand

Professor David Hand PortraitOne name kept coming up when we started our research into probability - and that was no coincidence.

David J. Hand OBE is a senior research investigator and emeritus professor of mathematics at Imperial College, London. He is also a non-executive director of the UK Statistics Authority and the author of The Improbability Principle.

Professor Hand’s work on probability has been recognised by publications including the Washington Post, Scientific American, New York Times and now, Online Bingo.

In the The Improbability Principle, he seeked to explain why coincidences, miracles and rare events happen every day’, using a unified framework of five simple laws.

He was the perfect person to speak to in our quest to discover why 99.5% of Britons are so bad at judging probability.

But first, let's get our laws in order.


  1. The Improbability Principle
  2. Mind over matter
  3. Interview with Professor David Hand
  4. - What influences our perceptions?
  5. - Evolution and odds
  6. - The effect of experience
  7. - The role of probability in gambling
  8. - Educating children in probability
  9. Further reading
  10. Summary

The Improbability Principle

Below are very brief summaries of the five laws of improbabilty. For a more detailed explanation of each, complete with real examples and a working demonstration, visit

1. The law of inevitabilility

The law of inevitability is based on the concept that something must happen. For example, if you bought tickets for every one of the millions of possible lottery number combinations, you would be guaranteed to hold the jackpot ticket.

You would have reduced your odds of winning from a massive 45 million to one, to a mathematical certainty. It would be inevitable. 

2. The law of truly large numbers

With over seven billion people on the planet, engaging in countless interactions and activities every day, it would be unthinkable for even the most far-fetched coincidences not to occur.

The law of truly large numbers suggests that 'with a large enough number of opportunities, any outrageous thing is likely to happen’.  

3. The law of selection

It’s slightly ironic that in hindsight, we can look back at unlikely events and attribute their wonder to the law of selection.

The law states that if you change the collection of things that you choose from, you can change the chances - you can make things as likely as you want if you choose after the event

4. The law of the probability lever

Have you ever been on holiday abroad and bumped into your neighbour from home?

Among the gasps of ‘what a small world!’, you would be better remembering that you both visited the same travel agent who happened to be pushing one particular hotel they had on offer -  the one that you are both now at.

The probability lever works as a metaphor for the influence of external or internal factors on the chances of an improbable event happening. Slight changes to a situation can make highly improbable events almost certain

5. The law of near enough

If you ordered a blue car and it was delivered in purple, would you be happy?

What if you ordered the car in blue and, unknowingly, a friend ordered the same car, but with a smaller engine and in purple. Would you wonder at the coincidence that you both chose the same?

You’ve just experienced the law of near enough. Events which are sufficiently similar are regarded as identical. 

Mind over matter

One of the less publicised chapters of The Improbability Principle relates to the impact of the human mind on our perception of probability and how people find it so difficult to behave in a random manner: ‘probability and chance often appear to be counterintuitive’.

Could it be these natural biases and idiosyncrasies that cause our judgement of odds to be so wrong?

We asked 1,500 people across Britain to rate the probability of six likely and unlikely events, in relation to the control event of dying in a road accident. Only 0.5% answered correctly.

We spoke to Professor David Hand about our results and their relationship to the various laws and principles covered in his book.

Read the full interview below. 

Horseshoes on wooden bench

What influences our perceptions?

OB: Our survey showed that 99.5% of people in Britain were unable to judge probability correctly. What do you believe are the key factors and influences that could affect a person’s perception of probability?

DH: I wasn’t surprised that the survey showed that so many people were unable to judge probabilities accurately. Mistakes typically occur when we are asked to judge something that we have no experience of or haven’t studied, but it also applies more generally to numbers in other contexts. There are many reasons for this, which include:

- A person’s lack of experience of the subject of the question;
- Our tendency to over-exaggerate small probabilities, the classic example being the lottery;
- A person's attitude to value and loss.

We’re particularly bad at judging very low probabilities - obviously because we will have very little experience of such events. Indeed, the only rational thing is to treat events with very low probabilities as impossible.

If something is so improbable that it’s unlikely to happen during your lifetime, or during the history of the universe, then it’s only sensible to treat it as impossible. This is called Borel’s Law. Winning the lottery is an example.

The chance that any particular person will win is so small it should be written off as impossible. But it isn’t because of people’s distorted impressions. One reason for the distorted impression is that people do still win the lottery every week.

What people fail to allow for is the fact that, while the chance of any one particular person winning is very small, if enough people play then it becomes likely that someone will win.

This is an example of the Law of Truly Large Numbers. And then the publicity raises people’s awareness of the possibility of winning, so distorting their impression of the probability of winning.

Borel's Law

Named after French mathematician and politician, Emilé Borel, Borel’s law states that ‘phenomena with very low probabilities do not occur’. It was originally intended as a rule of thumb for specific scenarios before they happen.

The law is an attempt to determine as precisely as possible which values of probability must be regarded as negligible under different circumstances. [1]

OB: Do you think this is an issue exclusive to Britain?

DH: The difficulty of estimating probabilities correctly isn’t confined to Britain. It's caused by universal biases of human beings. For example, one bias is that we are more confident than we should be in our estimates of probabilities. 

Culture can have an impact though. For example, if you ask people to measure their level of wellbeing out of 10 in the West, you’ll get a basic level of around seven. In some other countries this basic level is lower. It’s influenced by cultures, religions and ways of life, and all of these can also affect a person’s perception of chance. 

OB: Does the intense media focus on rare events offset people’s prior perception of their rarity to such an extent that they believe something to be more likely than it actually is?

DH: The impact of the media is quite large and does change people’s perception of risk. I suppose one of the best examples of distorted impression is the period after 9/11. People stopped flying because of fear that a plane might be brought down, and switched to cars instead.

But this caused more deaths on the roads. Flying is statistically a much safer form of travel but people’s irrational fear moved them out of planes and into cars, from a safer to a more dangerous situation. 

Tiger hiding in grass

Evolution and odds

OB: Do you believe the various cognitive biases that cause us to overestimate chance and probability are related in any way to physiological reactions such as fight or flight? Could they be a legacy of human evolution?

DH: Absolutely. I think this is spot on. Our failures to realistically assess probabilities are certainly a consequence of our evolutionary heritage. One point is that it’s not just the chance that something will happen that we have to consider, but also the severity of the consequences.

Even if I think the chance is tiny that the movement in the grass means there’s a tiger there, the consequence of it being a tiger is so great that I will overestimate the chance that there’s one there.

OB: In light of this, is there any evidence that animals share the same biases of humans? 

DH: There are studies which show that animals share these biases. Perhaps this is further evidence of its evolutionary nature. Animals can be conditioned to behave in certain ways when presented with appropriate stimuli - for example, feeding them after ringing a bell.

Having learnt this behaviour they continue to exhibit it, even if the circumstances change - ie. the ringing of the bell will continue to make them salivate, even if food is no longer delivered. They have acquired a superstition, like the belief that a black cat crossing your path will bring bad luck.

The effect of experience

OB: Our survey showed that younger people are better at judging chance and probability. Why do you think this is? And, in contrast, why are older people are worse?

DH: I was surprised to learn that about younger people. However, older people are more risk-averse due to the fact they have more invested in life, such as property, children, family and wealth. Being risk-averse can cause you to exaggerate the probabilities of dangerous things.

There is also the current experience vs recency effect. If you have recently been exposed to some phenomenon, you’ll likely have a better understanding of it. Dying in a road accident is something that younger people might be able to relate to better as they might be on the road more often.

Older people may travel less frequently and therefore may be more fearful of road deaths as a result.

OB: Does the positivity/negativity of an event affect how people perceive its likeliness to happen?

DH: Studies have shown that people are more averse to losing something than they are gaining something equivalent. You would probably be more upset about having £1000 taken away from you than you would be pleased about having £1000 given to you.

Losses and gains have to be put in context. It’s irrational for a poor person to spend £2 on a lottery ticket, but rational for a billionaire to do so.

Also, I think it’s important to recognise that when you buy a lottery ticket, you are buying a dream. You’re buying an experience - analogous to buying a cinema or concert ticket.

Dice and casino chips

The role of probability in gambling

OB: Do you believe gambling companies actively use media and marketing messages to influence how players perceive probability?

DH: It is certainly the case that gambling companies use advertising in this way to lure people into playing, but it’s necessary to recognise that they are a business. They offer a service that people want to buy. From a company perspective it makes sense to imply that a player has a good chance of winning.

It’s the business of lottery companies to sell lottery tickets, and one way they can do this is to avoid people getting bored with the games by frequently creating new games and scratch cards. But in terms of the messaging actually influencing the perceptions of probabilities, I’m not sure.

There is a shared responsibility in gambling. The company has a responsibility - rather like drinking too much and the barman saying you’ve had enough. But on the other hand, people ought to know how to behave and understand their own limits. It’s their responsibility as well.

Having said that, though, obviously there will always be vulnerable people, and this is where the line starts to blur. 

OB: To what extent do you believe poor perception of probability could be responsible for a player’s addiction to gambling?

DH: This could well be the case. There will always be some people who spend more than they have or can afford. I can see definite links between a misunderstanding of probability and problem gambling.

One familiar example is the so-called gambler’s fallacy. This is the notion that, if a fair coin comes up heads ten times in a row then it’s more likely to come up tails next time. But this cannot be true - after all, the coin does not have a memory.

The Gambler's Fallacy

The gambler's fallacy is the mistaken belief that, if something happens more frequently than normal during some period, it will happen less frequently in the future, or that, if something happens less frequently than normal during some period, it will happen more frequently in the future (presumably as a means of balancing nature).

This fallacy can arise in many practical situations although it is most strongly associated with gambling where such mistakes are common among players. [Source: Wikipedia]

One of the most famous examples of the gambler's fallacy in action was in August 1913 at the Casino de Monte-Carlo in Monaco, where the roulette ball landed on black 26 times in a row! In the mistaken belief that the wheel had to correct the imbalance, players continued to bet red and subsequently lost millions. This remain the longest streak of a single colour ever recorded at odds of 136,823,184 to 1.

OB: Recent regulation changes have ensured that gambling sites are more transparent in their communication of odds, chance and success to players. What impact do you think this could have on the gambling industry as a whole?

DH: I think better communication could have some impact, but still believe that people would often overlook the odds. I think this is indicated by the lottery. The chance of winning the jackpot is less than that of getting 25 consecutive heads in tosses of a fair coin. But people still buy lottery tickets even though they know the chance of winning is very small.

OB: At what point does a positive event (something that provides pleasure) become a negative one and at what point do people become too invested to give up?

For example, you refer to people buying a dream when they buy a lottery ticket. That may be the case for a casual player, but for somebody investing £100s in tickets every month, does the experience become less fun and more desperate? Are they no longer chasing the dream but chasing their losses?

DH: A nice question. You are describing a classic problem gambler behaviour (unless they are billionaires, to whom a few hundred pounds is inconsequential). As to chasing their losses, this is the classic sunk cost fallacy of economics.

The point about the lottery, and many other gambling situations, is that each game is fresh, starting from the beginning again. The rational thing to do is to accept that the money has been lost and move on.

OB: The hot hand fallacy provides a previously successful person with an illusion of control over a random event. Is there such a thing as a lucky streak or is this simply the mind overestimating its ability to overcome probability? What are the dangers and opportunities this poses?

DH: This is a little controversial, but studies with real data, rather than people’s perceptions, show that the hot hand fallacy is just that - an illusion. It is, as you nicely put it, simply the mind overestimating its ability to overcome probability.

The mind plays many tricks with probability. I recall a friend proudly telling me about his big wins betting on horse races. But talking to his girlfriend revealed the very large number of losing bets, which he had (doubtless subconsciously) suppressed.

Abacus educating probability

Educating children in probability

OB: Do you believe that schoolchildren are sufficiently educated in probability given that by the age of 18 they will be legally able to gamble, smoke and drive?

DH: No, I don’t think they are. Getting people to have a gut feeling for chance can be done much better than it is at the moment. People aren’t getting enough maths education in general.

Numeracy and quantitative skills are essential and influence many decisions in adult life. And they are becoming more and more important in the modern world. A sound grasp of chance and probability is increasingly important.

However, as your survey shows, there are may well be cases where young people’s perception is strong. Road accidents might be an example. But this is a specific event where they have direct experience and are particularly vulnerable.

Young people may have friends or know classmates who have died in road accidents. It is less likely that they have experienced a friend dying of lung cancer as a result of smoking - certainly not within their own generation - so their understanding of the probabilities associated with smoking could be less accurate.

Legal ages in the UK

  • Age 16 - Play the National Lottery / Give consent to sexual activity
  • Age 17 - Drive a car / Become a blood donor / Apply for a private pilot's licence
  • Age 18 - Place a bet / Buy and drink alcohol in a bar / Buy fireworks / Buy cigarettes / Get a tattoo / Vote in local and general elections

Sources: The Mix / Law Stuff

OB: How do you believe people's behaviour would be affected by a greater understanding of the laws of probability?

DH: I think that raising awareness of probability could affect how open people are to creating fortunate circumstances and luck. There are classic studies along these lines, showing that outgoing people tend to be luckier.

In fact, of course, they are simply putting themselves in more situations and giving themselves more opportunities for good things to happen. Understanding how you can affect the odds can be empowering and motivational.


David Hand is undoubtedly one of the country’s top thinkers in the world of probability and statistics.

His insights into our survey results and general human nature not only offer a window into the workings of our own minds, but also allow us to better understand the reasoning behind our most common assumptions.

The implications of our poor perception can range from being as trivial as losing a coin toss to as critical as driving more safely.

It's time to tackle our probability problem, head (or tails) on.

Let us know what you think of Professor Hand’s answers in the comments below. 

We would like to extend a huge thank you to David Hand for generously taking the time to speak with us. 

More from Professor David Hand

Follow the links below to find out more about David Hand and The Improbability Principle.

More on luck and probability

Discover the other articles in our luck content series and learn about the real-life situations that prove David Hand's theories.

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