Misconceptions about probability theory and statistics have major repercussions on society. From seemingly minor things like the excessive sensationalism of some headlines, all the way to the jailing of innocent people based on “statistical evidence”. One of the most common misconceptions is the so called Gambler’s fallacy. Wikipedia defines it as follows:

The gambler’s fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the belief that if deviations from expected behavior are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future.

This definition may seem a bit abstract, so let’s clarify it through a practical example. What’s the probability of flipping a fair coin 10 times in a row and obtaining heads consecutively each time? The answer is:

[tex]\displaystyle \mathrm{P(E)} = (\frac{1}{2})^{10} \approx 0.0009766[/tex].

This would be very unlikely. How unlikely? One in 1,024 to be exact. So if we’ve just observed the coin appear as heads 9 times in a row, what are the odds that the same coin will land on heads on the 10th toss?

Many people would argue that the chance of this happening is less than one in a thousand, as we just calculated. However, that answer is blatantly wrong. The probability that the 10th fair coin toss is going to come up as heads is still 0.5, because each trial (toss) is statistically independent from those that preceded it. Tossing 9 heads in a row is very unlikely, however once it has happened, it doesn’t influence the outcome of the 10th toss in any way.

People who fall for this fallacy, do so because of a fundamental misunderstanding of how probability works. They combine the probability of past events (irrelevant for independent trials), with that of future events. With the example above, some people would also erroneously conclude that “tails is long due to come up” and as such would think that it’s more likely to occur.

It’s not a difficult theory to understand, but a lot of people make the mistake of confusing probability with sheer luck. Every instance of an event relies on the same probability regardless, whether you’re rolling dice, tossing a coin, or even waiting for buses. If the odds were 5000:1, 4999 events later you’ve still only got a 0.02% chance of the odds going in your favour, the same as the first time the event occurred.

This informal fallacy has contributed to the ruin of many gamblers over the years. A tragic example of what happens when you uphold this way of looking at odds occurs with many who play the game of “Lotto” in Italy, a very popular lottery game played amongst the general population.

The idea behind this game is very simple. Five distinct numbers between 1 and 90 are randomly selected in 10 different Italian cities, three times a week. Gamblers can place several types of bets, but the one we’re interested in, for the sake of this article, is called the “estratto semplice” (simple draw). This type of game requires gamblers to correctly predict that a specific number will be drawn in a particular city.

The probability of placing a winning bet is 1 in 18 (i.e., 5/90), while the payout is 11.232 times the amount that you put down (so if you bet 1 Euro and won, you’d walk away with 11.23 Euros before taxes). The odds are clearly stacked in favor of the house, of course. Incidentally, Lotto is run by the state and is also known as “a tax on the stupid” for rather obvious reasons.

There are many “systems” and theories used by a large pool of gamblers who want to “beat the system”. More often then not such systems are based on some flawed understanding of how probability really works. A very popular theory is that of the “numeri ritardatari” (“late numbers”, as we will refer to them throughout this article). The basic principle behind late numbers is this: since it’s extremely unlikely that a given number will fail to appear at least once out of 150, 180 or 200 draws in a row, in a given city, you can identify what numbers are “due” to appear and thus bet on them. For example, if a number hasn’t been drawn in the past 140 trials, the number of bets on it will start to grow very quickly.

Of course, despite the fact that a number hasn’t come up in a given city 140 times in a row, its probability of occurring on the next draw is still just 1 in 18. So betting any of the other 89 numbers would yield the same probability of winning.

The application of this fallacy becomes extremely dangerous when coupled with Martingale betting systems, which are often adopted by “late number theorists”. The theory they use is very simple. Since they assume these late numbers are “due” very soon, they think they are going to be able to afford to put down double their previous wager on every bet until the number eventually appears. So when it does happen, the last sum they bet is multiplied 11 times (for the payout) and they will recoup all the money they’ve spent up until then, and end up netting a large additional payout, which is the (last wager x 9.232 + 1) Euros.

Martingale betting systems are guaranteed to work provided that the gambler has an infinite amount of capital and no limits are imposed on the maximum bet that’s allowed to be placed. In the real world, both of these requirements cannot be realistically met. The amount bet grows exponentially, so the Martingale system ends up being a surefire way to bankrupt those who employ it.

In the case of the Italian Lotto, both the fallacy that late numbers are “due” and the choice of betting systems (Martingale) are responsible for the ruin of many. The gambler’s fallacy plays an important role in this case because most people realize that they can’t sustain a Martingale type system for 200 consecutive draws. It’s their faith in the idea that late numbers are very likely to pop up soon, that tempts them into toying with this risky system.

If we assume these people are convinced that a very late number (say, one that hadn’t been drawn in the past 180 lottery draws) will be selected at some point during the next 5 weeks or so (15 trials), and that they’re starting with a bet of one Euro, we can see that the maximum amount they’d need to invest (according to their theory) would be 32,768 Euros, with a max bet of 16,384 Euros by the 15th draw. This is a sizable sum of money, but something that some people would still be able to put down, especially because they knew they payout would be 184,025.088 Euros (before taxes). A tempting prize indeed.

But what are the real odds that the number in question, the one that’s been eluding the gamblers, will not end up occurring at least once in the next 15 draws?

[tex]\displaystyle \mathrm{P(\overline{E})} = (\frac{17}{18})^{15} \approx 0.4243[/tex]

So there is a 42.43% risk that the punter will lose their 32,768 Euros, because they won’t have sufficient funds to double their wager at the next turn (assuming 32,768 Euros was the maximum amount they can afford to bet).

Bear in mind that with an exponential growth of the bet, a huge amount of capital will only afford our late number gamblers a few extra draws, thereby only slightly increasing their probability of making a profit. (With a payout of 11.232 times the wager, they could afford a smaller increase in the amount of money they put down draw by draw, but the overall principle remains the same.)

What has an adoption of this faulty theory led to in Italy? What kind of impact has it really had on those who adhere to it? The honest truth is that it’s gone so far as to contribute directly to things like suicides, people swindling their friends and employers, divorces, people betting their life savings and their homes, families being destroyed, and so on. Do such dire consequences occur to everyone who plays this game? No, of course not, but the fact that it’s happened to some people, and that these flawed theories are still employed today, is indicative of the misunderstanding about probability (and the risks of gambling) that occurs in the general population.

One could – and should – argue that such peoples’ demise is due to their gambling habits and to good old fashioned greed, yet I can’t help but feel that a solid understanding of probability theory would go a tremendous way in helping to cut down on the number of people who fall prey to these types of widespread theories.

An increased awareness of probability and statistics can only improve society and its ability to assess situations and make rational decisions. How do we begin to remedy this situation, not only in Italy, but around the world? We can start by devoting far more time in grade, middle and high school math classes, in order to teach students about this important subject and the implications that it can have on their everyday lives, understanding of society, and ability to make wise financial decisions.

If I saw a coin appear as heads 29 times in a row, I’d think that it’s quite likely that the next flip will also land on heads.

Because I’m stupid with statistics? No. Because I’m smart enough to account for the difference between theory and practice: something is wrong with your coin. Maybe it’s not quite as symmetrical as you think it is.

The probability for a single launch of coin to see “head” is 0,50… ok.

I see a difference probability for 29 consecutives launch heads and for 30 consecutives heads, or not?

Hehe, Alain. I see where you are coming from. However, this is a fair coin and by definition there are no issues with it. If it helps, reduce the number of tosses to 5, to make it a “less suspicious” coin for you.

I think the both of you are right at the same time, but you are talking about 2 different problems :

Alain is talking about statistical inference, where the probability associated to our model (toasting a coin)is unknown and it would be rather irrational to ignore such indicator as a 29 consecutive heads a row in the absence of counter example.

In the other hand, Antonio is taking about Probability theory, or how to use ( in an unbiased manner) a well defined probability (and not an estimation)

I tend to agree with Alain. While this blog is right about “in theory…” but in reality, there are many variables that maybe we don’t even know we’re supposed to account for.

In fact, perhaps the opposite of the “gambler’s fallacy” is correct. If the numbers are trending away from the “expected average,” maybe something is wrong in your computation of the expected value (maybe not, but maybe so).

It would make more sense to me to play numbers that HAVE come up recently since you know that it’s at least possible for those numbers to come up…maybe the others have something working against them (an error in the random-number-generator, the balls in the machine are weighted funny, etc.).

This reminds me of the hindsight cognitive bias, as it’s related. A classic example goes somewhat like this: two separated juries must decide if a city group was negligent for not hiring a bridge manager in case of flooding caused by blockage of the bridge. They’re given all the data the city had when making its decision, and then one group is also given the past history that such an event has actually happened. The group with that little extra detail gave consistently higher estimates for the probability of flooding and need to hire a manager (or further support the bridge) than the group who didn’t know about it. Also interesting is that telling juries to avoid this bias doesn’t help it at all.

So I agree generally that a better general understanding of probability theory is important and could hep the problems of various biases and fallacies. In the math education system though people wince when they hear they’re going to do probability, and a lot of teachers just hand-waive over Bayes’ Theorem (http://yudkowsky.net/rational/bayes) instead of actually dissecting it to see how important it is.

Sorry, Antonio. Didn’t mean to make you change it. Of course you’re right (with 10 or with 30) in an IDEAL world. I think Alain and I were just saying that in “real world” situations, things may not be as ideal as they seem.

And your deeper point about people thinking things are “due” does need to be addressed.

When asked about his multi-billion dollar investment strategy Taleb said “Odds are the coin will come up heads. Why, because someone is cheating.”

Guys, I changed the example from 30 to 10 tosses, so that we avoid “missing the forest for the trees”.

For your calculation of the additional payout for martingale, i think the correct formula should have been

(last wager * 9.232 ) + (initial wager)

Sudeep, the 11.232x payout includes your last bet, and the initial wager is assumed to be 1 Euro here. The general formula is: (last wager * 8.232) + initial wager.

Can you suggest some books where one can gain a better understanding of probability and statistics?

Dhinesh, there are many valuable books on the subject, depending on your mathematical background and approach to learning.

That said, an excellent textbook that is not too math-intensive is:

Statistics, 4th Edition by David Freedman et al.

People also miss the point that a spread in the frequency with which numbers will be drawn is to be expected in a purely random process. Which number will be more rare is not predictable, and the future is not predictable. Only the fact that there will be a spread in frequencies is predictable. If the frequency of draws if the individual numbers is “too even” it implies a process that is not random.

I read that it is suspected that Mendel (the first geneticist) “improved” his results–results close to 75% to 25% appearance rates of characteristics occur “too often” in his data. In his time, the statistical methods to check for this were not yet developed.

Antonio, this is an excellent article, so I wanted to say thank you. And echo another call for suggestions on books on statistics/probability for non-statisticians.

Alain, you have a good point. If I saw a coin come up heads 29 times in a row, I would tend to think the coin were biased/double-headed/someone was cheating.

With that said, if I knew with certainty that the coin was fair then it would certainly be possible for it to come up 29 times in a row. In fact, if memory serves someone once proved that for any n you will eventually get n heads in a row if you keep flipping long enough, its just a question of how long that will take. And once you got to that 29th head with the fair coin (unlikely, but possible) then the next toss is 50-50 heads.

Another good fallacy is “completing the problem”.

If I flip two coins simultaneously, then show you that one is heads… what is the probability that the other is heads?

Most people will say “50/50”, since they’ve heard that each coin flip is independent, or “1 in 4”, since that’s the chances that 2 heads were flipped originally. That’s not really the case here. There are 4 possible outcomes of the 2 coins flipped, but I showed you one was heads, which eliminates the tails-tails outcome. Of the 3 remaining possibilities, only one is heads/heads, so the probability that the 2nd coin is heads GIVEN that the first coin was heads is 33%.

I do a good rundown of Keno odds on my blog:

http://www.discreteideas.com/2009/07/keno-an-odd-game/

(continued) Basically, I think we’re discussing Bayesian probability here.

My prior probability that the coin is fair (without seeing a flip) is 0.5. Observing a run of heads that occurs more frequently than explained by randomness decreases my confidence in the fairness of the coin – hence, the posterior probability of the coin being fair is lower. Likewise, observing runs that fall within the expected distribution of runs increases my confidence in the fairness.

BUT assuming that the coin IS fair, the probability that the coin will come up heads has not changed. It is the uncertainty about the state of the coin that is the key here.

(Of course, that doesn’t explain why people believe seeing a 30th head after the 29th is so unlikely. If the coin was actually unfair, the odds of seeing a 30th head are actually quite good [much higher than 0.5]. Bet against the crowd, I guess.)

Re. The coin coming up heads 29 times.

Nassim Nicholas Taleb has an excellent example in “The Black Swan” about this. But for now:

@The Count:

Your example is nice, but it is not a fallacy. The problem by flipping 2 coins is defining the probability space. Different definitions yield different probabilities for the event that the 2nd coin is heads.

@Alain:

That’s why goodness-of-fit tests exist. If the deviation from the uniform distribution is significant, then we can assume that something is wrong with the coin.

hi there antonio,

this might not pertain to the article, but i saw that you probably have an italian background.

i’m trying to decide wether to enroll in the facolta’ di matematica or in chimica in the university of Pavia?

did you study in italy? what can you tell me about mathematics?

thanks

@The Count:

Your example is nice, but it is not a fallacy. The problem by flipping 2 coins is defining the probability space. Different definitions yield different probabilities for the event that the 2nd coin is heads.

@Alain:

That’s why goodness-of-fit tests exist. If the deviation from the uniform distribution is significant, then we can assume that something is wrong with the coin.

Sandeep wrote:

>

>For your calculation of the additional

>payout for martingale, i think the correct

>formula should have been

>(last wager * 9.232 ) + (initial wager)

>

>

Antonio wrote:

>Sudeep, the 11.232x payout includes your >last bet, and the initial wager is assumed >to be 1 Euro here. The general formula is: >(last wager * 8.232) + initial wager.

Say that 5 bets are made and then the 6th one is a win.

First bet: bets 1 loses 1

Second bet: bets 2 loses 2

Third bet: bets 4 loses 4

Fourth bet: bets 8 loses 8

Fifth bet: bets 16 loses 16

he loses a total of 31 before he wins.

When he wins, he wins 32*11.232 .

So the net is:

won = 32*11.232

lost = 31

net = won – lost

net = 32*11.232 – 31

net = last_wager*11.232 – (last_wager-1)

= last_wager*10.232 + 1

This seems very confusing to me since it isn’t either of the two other answers.

I originally thought I might have the same answer as Sudeep. My understanding is that by 11.232 what is meant is that if the bettor has only 5 euros in his pocket and he bets all 5 euros then if he wins he can put 11.232*5 = 51.16 euros in his pocket.

I see though that your answer to Sudeep and your formula in the blog post is consistent:

answer to Sudeep:

(last wager * 8.232) + initial wager

blog post:

(last wager x 8.232 + 1)

given that initial wager=1

> Say that 5 bets are made and then the 6th one is a win.

#1: spends 1, wins 0

#2: spends 2, wins 0

#3: spends 4, wins 0

#4: spends 8, wins 0

#5: spends 16, wins 0

#6: spends 32, wins 32*11.232

So the net is:

won = 32*11.232 = 359.392

spent = 63 (1 + 2 + 4 + …)

net = won – spent = last_wager * 11.232 – (2*last_wager -1) = last_wager * 9.232 + 1

Sudeep’s formula is the correct one. I updated my post.

Thanks for an interesting read. I knew about most of these, but not the specifics of the Italian Lotto. Another source of misunderstanding probability occurs when humans try to do mental calculations involving very large numbers. A million is about the same as a billion, right? 🙂

“If I flip two coins simultaneously, then show you that one is heads… what is the probability that the other is heads?

Most people will say “50/50″, since they’ve heard that each coin flip is independent, or “1 in 4″, since that’s the chances that 2 heads were flipped originally. That’s not really the case here. There are 4 possible outcomes of the 2 coins flipped, but I showed you one was heads, which eliminates the tails-tails outcome. Of the 3 remaining possibilities, only one is heads/heads, so the probability that the 2nd coin is heads GIVEN that the first coin was heads is 33%.”

This is incorrect.

Ther are four possible outcomes:

1.Heads-Heads

2.Heads-Tails

3.Tails-Tails

4.Tails-Heads

By showing me the first coin is heads, you’ve eliminated TWO outcomes.(#3 and #4)

So the probability the other coin is heads is still 50-50.

So let me put this another way. A betting shop has determined a total for points scored in a football game and is taking bets on if you think the actual total will be under or over their number. Over the last 1000 games the bet shop’s numbers are below the actual total 50% of the time so of course they are also over the actual 50% of the time. It’s a 50/50 bet.

I will take a coin and designate heads as “over’ and tails as under, then flip it and and compare it to the outcome of the game. (the hidden coin)

There are four possible outcomes:

Over-Over

Over-Under

Under-Over

Under-Under

I flip the coin and its heads or “over”

so by your rule I have eliminated one outcome, the “under-under” and three outcomes remain. So again by your rule I have only a 33% chance the game will be “over”.

What a great idea! Before I bet I’ll just flip a coin and choose the opposite of the result and I’ll have a 66.67% chance of winning a 50/50 bet.

I’ll send you a postcard from my Villa in France.

This all sounds extremely similar to some of the rhetorical fallacies you learn in english class. Though, admittedly, there’s more data behind this.

Dear enthusiastic math lover..after seeing comments above, i think i want to contribute a little here.

@alain: yes, real world problems are complex. That is why theories are developed: to simplify problems n get the solution of it. As stated above,we ASSUME tht the dice is fair. Then,from such assumption,we have the theory of measuring the probability to support our decision. If you consider that the assumption is not valid in real world problem you face,e.g. the dice is not fair,you may consider not to employing the theory. But then what happened? We may end up dealing with bigger complexity.. So,we may as well put things in the proper corridor in order to get the best theory we can employ to support our decision.

@rick: i believe you assume that the sequence of the outcome is not necessary. Taking such assumption into account,you should have known that the outcome of the coins are:

>over-over

>under-under

>over-under.

Just 3 instead of 4 since you consider that {over-under}={under-over} without realizing it. By knowing that the outcome of one coin is ‘over’, Therefore we have eliminated the outcome {under-under} and thus the options we have left are {over-under} n {over-over} which yields the probability 0.5 of having {over-over} as the outcome.. I hope this is comprehensible. *sigh,typing this comment via cellphone is surely tiring..

Be sure to check math.web.id to see the various colour of math undergraduate in blogging.merci.

Hello,

What can I have to do to provide a version of this text in my blog? I am willing to post it in Brazilian Portuguese

thanks

Leonard

Hi Leonardo,

simply state that it’s your translation from the original called “The Cost of Not Understanding Probability Theory”. Link to it. And you are good. 🙂

In all the years experience in online betting I encountered many people claiming to Martingale you can get rich. Yes, you can, but if you limit the maximum bet and has a large bank, all of which are mutually exclusive nowadays in all online and offline bookmakers or organizers of gambling games.

@Rick: Hehe, it would be very easy and good to be true.

@Edward is right that behind these calculations are unambiguous figures. But the fact is that mass people betting in bookmakers have not touched the magic never called mathematics and from there just gamble driven by the psychological element and not guided by logic. The easiest way to win is not to play something like the advice of Dale Carnegie – The easiest way to win an argument is to avoid it.