One of the core concepts in my book, The Skeptics’ Guide to the Universe, is that humans are inherently good at certain cognitive tasks, and inherently bad at others. Further, our cognitive processes are biased in many ways and we tend to commit common errors in logic and mental short-cuts that are not strictly valid. The human brain appears to be optimized by evolution to quickly and efficiently do the things we need to do to stay alive and procreate, and this has a higher priority than having an accurate perception and understanding of reality. (Having an accurate perception of reality has some priority, just not as much as efficiency, internal consistency, and pragmatism, apparently.)

One of the things humans are not generally good at is statistics, especially when dealing with large numbers. We have a “math module” in our brains that is pretty good at certain things, such as dealing with small numbers, making comparisons, and doing simple operations. However, for most people we quickly get out of our intuitive comfort zone when dealing with large numbers or complex operations. There is, of course, also a lot of variation here.

We give several examples to illustrate how people generally have poor intuition for statistics and certain kinds of math, and how our understanding of math runs up against our cognitive biases and flawed heuristics. These common examples include the fact that we have a poor intuitive grasp of randomness.

Probability also seems to be a challenge. How many people would you have to have in a room before having a >50% chance that two of them share the same birthday (not year, just day)? The answer is a lot less than most people guess – it’s just 23. We tend to underestimate how probabilities multiply when making multiple comparisons. This is why we are inappropriately amazed at coincidences. They are not as amazing as we naively think. The probability of someone winning the lottery twice is also a lot higher than you might think.

Another favorite example is the Monty Hall problem. Quickly – if in the famous game show, Let’s Make a Deal, Monty Hall shows you three doors. One door hides a car, while the other two hide goats. Monty Hall know which one hides the car. You pick a door, and then Monty opens up one of the two doors you did not pick to reveal a goat (he always reveals a goat – remember, he knows where the car is). Monty then offers you the chance to change your pick to the other unopened door. Should you do it? The answer is yes. You will increase your odds of winning from 1/3 to 2/3. Many people have a hard time wrapping their head around the statistics, and insist your chance of winning is 50-50 regardless, but that is clearly wrong.

This all brings me to the primary example I want to discuss today – the gambler’s fallacy. I received the following e-mail question, again demonstrating how counter-intuitive statistics can be:

On page 89 you talk about tossing a coin; the independence of each coin toss and the bias in thinking that after 5 heads in a row a tail must be due – the gambler’s fallacy. This I feel I understand very clearly, obviously a previous toss can have no influence on a subsequent toss. However, and please point out where my thinking is wrong, I feel there is another factor which may mean that it is reasonable to expect a tail after a number of consecutive heads. I will attempt to set out my thinking in a logical way…

If a process where there are two possible outcomes (A and B) is repeated multiple times (let’s say 1000) and the results are recorded, it is reasonable to assume that the number of times there will be no A’s recorded will tend towards 0%, and 1000 A’s will tend towards 0%. Therefore the number of times A is recorded 500 times will tend towards 100%. (I sketched out a graph but I can’t add it here). Is it not reasonable to assume, therefore, that after a run of all A’s one ought to expect the likelihood of a B to be higher if the tendency to end up with half A and half B is to be maintained. I hope you follow my thinking.

This is a very common cognitive error, and can be very subtle. In fact, the one mistake I am aware of in my Teaching Company course on critical thinking contains this very error (dang). Try to see if you can detect the e-mailer’s error.

I think there are a few ways you can think about this. First, it should be enough to simply point out that each fair toss of a coin is an independent event with a 50% chance of either heads or tails. Nothing that happens before has any influence on this probability (again – taking the premise that it is a fair coin and a fair toss). The e-mailer knows this but is having a hard time squaring this fact with the other fact that statistically the more times you toss a coin the more the numbers of heads vs tails should tend toward 50%.

The tendency toward 50% heads and tails, however, is purely statistical. The numbers will take a random “drunken walk” and will simply tend to average out over many trials.

But here’s the thing – this is true no matter when you start – meaning from that point forward the statistics will also tend toward 50% of each. The primary error is that the e-mailer is not aware that by starting “after a run of A’s” he is biasing the sample. He is essentially cherry picking his starting point. But – from that starting point forward, the coin flips will tend toward 50% each. The universe will not conspire to make subsequent tosses compensate for the prior run of A’s – which the e-mailer has cherry picked by the very premise of the example.

Hopefully, after being explained, this should seem obvious. But – what’s interesting is that this very logical fallacy crops up in actual scientific research all the time. The most blatant way, popularized by ESP researchers, is called “optional starting and stopping.” Let’s say they are looking at a sequence of trials of psychic ability, such as guessing the toss of a coin, or reading the Zener cards. Each trial is either a hit or a miss, so is binary like a coin toss. ESP researchers in the past, hoping to demonstrate psychic ability in the data, came up with the technique they called optional starting and stopping – look at a long sequence of data and then start counting at the beginning of a run of hits, and stop counting before a run of misses. This was justified by saying that psychics need a warm up period, and then they get fatigued.

However, hopefully it is now obvious that they are just biasing the sample they are using, and are essentially manufacturing positive results. This is why in real science you have to count every data point, and you cannot pick and choose.

But even still this fallacy crops up in more subtle ways. Let’s say a researcher notices a correlation between A and B, so they run statistics and find that A and B do correlate in a way that deviates from pure chance (the equivalent of having 60% heads and 40% tails). But the data is limited, and this could just be a fluke, so they gather more data. However, they include the original data in the final data set, run the statistics, and find there is still a correlation (but maybe it’s a bit smaller). What was their error?

When gathering data to confirm an apparent effect or correlation, you should not include the original observations. If you do, then you are carrying a quirky random fluke forward in your data, and you are not independently confirming it. To give an analogy, it’s like looking at a string of heads vs tails, seeing a run of tails and hypothesizing that the coin is not fair. You then do 100 trials of tossing the coin – but you also include the original string of tails in the data. So essentially you just cherry picked a random clump of tails and then carried that forward in the data, biasing the sample.

This may not feel wrong, because we intuitively underestimate the probability of noticing a random clump of correlations out in the real world. We may also fail to appreciate the fact that you have to have a random or fresh starting point for counting data in order for the statistics to work. You cannot include any data that is biased in any way – even by the fact that you happened to notice a clump in the randomness.

Our poor intuitive grasp of such statistics affects our understanding of the world in our everyday observations, but also in the way we conduct science – which should be obsessively rigorous. It is, when it works properly, but often even scientists make these kinds of mistakes. Sometimes entire fields of dubious research are based on such cognitive errors. The Gambler’s Fallacy lives on.