Deceptions Of Probability

9/24/2026

Considering my mathematics career since high school, probability and statistics always caused me troubles. Although I did very well in the other areas of math, – and all the other things in world, of course – those colored balls and flipped coins never seemed pretty or useful to me. British Prime Minister Benjamin Disraeli (1804 – 1881) said that: “There are three kinds of lies: lies, damned lies, and statistics”. For sure, only humans lie, so the problem is not in science, but in us. I have noticed how we deceive others and ourselves in several ways in the areas of probability & statistics. Let’s review some of them, so we will all remember high school exams once more.

Sample Space

One day, we were driving on peaceful (!) streets of Istanbul with a friend. When talking about the durability of cars, he told me that: “Before buying something, look for a brand’s old versions on street. For instance, there you can come across an old Volkswagen Polo frequently. Why? Because they are durable, and people still use them. You don’t see any old Peugeot 206, do you? Why, because they did not endure and people sold them to junk.” At first, that seemed like a bright observation to me. But the reasoning had a very simple flaw: two observed quantities do not have the same sample space, so they can not be compared. That is: Let’s define the old model of a car as the model that is older than 10 years. Let’s assume that, on an ordinary cruise we have encountered 5 old Polo, and 1 old P206, and observation is more or less repeatable. Actually two quantities are not comparable by themselves, their sample space is not common. Assume that, before the previous decade, 6.000 Polo’s we sold in Istanbul. However, for the same period let’s assume only 1000 P206’s we sold. So, if remaining conditions are same, 1 P206 against 5 Polos is not a surprise. In fact, 6 Polos for 1 P206 would not be surprise either!

Psychology of Observation

Sometimes we have the correct sample space, and we make the deductions in a correct way. But we do not gather information correctly. Correctness is a virtue of computers but not humans. So we are more likely to record the data that we prefer to record.

NBA fans have the phrase of “hot hand” in their jargon. For instance, Kobe Bryant takes the ball and makes a three-pointer. Just 30 seconds later, Kobe makes the second three-pointer in a fast break. Wow, now he has the “hot hand”, and a minute later, Kobe tries the third three-pointer and scores it! He is more likely to score, if he tries the fourth one immediately (when his hand is hot)… Is it?

It isn’t actually. Statistics clearly show that, players tend to shoot a lower percentage after making shots than after missing them.[1] Thus, if Kobe scored 3 three-pointer shots in row, it is definitely more likely to for him to miss the fourth one. But why are the fans are mistaken? That is very understandable, since it is really something exciting if some player has a hot hand. It is something to be appreciated, something to be told. People do not remember alternating scores and misses, instead they remember good deeds, consecutive scorings.

Psychology of observation always affects the reasoning of people, and sometimes may lead to illogical or mystic beliefs. It is common that people claim, when they see something particular (a crow for instance) in their dreams, someone around them dies or something bad happens. This event might have happened for sure, but it is important to question how many times has the claim owner seen a crow in his/her dream? Perhaps he/she dreams a lot, and he/she is much likely to remember “saw crow – something happened” cases than “saw crow – nothing happened” cases.

Dismissing Other Variables

World is not simple, even the events that seems to be under strict control may be affected by many conditions and parameters. However, people usually tend to think simple and straight. Sometimes we observe and measure things correctly, and still make false deductions, since we have not measured or we are not aware of the real parameters that are affecting the outcomes. We always hunt the easy evidence, and sometimes miss the real stuff. Dan Ariely, in his popular book Predictably Irrational, provides a nice example about this phenomenon.[2]

Mr. Ariely worked with Israeli Air Force in order to increase the effectiveness of pilot training. At some time, he witnessed the punishment of student pilots in case of bad performance. Nearly everyday, student pilots had their practices on air, and do several tasks. If they fail or score bad, then they are heavily berated by their chiefs in front of their fellows. Mr. Ariely asked chiefs’ opinion about punishment, and they defended themselves : “it is useful, guys do it better after punishment”. Students also agreed about the benefit.

However, Mr. Ariely is not convinced, and decided to conduct an experiment. For students in three different classes, three methods were applied. For class 1 students, after a failure, students were punished as before. For class 2 students, after a failure, students were heavily encouraged like “it is ok, you will do better, you have the potential”. And for class 3 students, after a failure, nothing is done.

Results were surprising. It is seen that success trend for class 3 (nothing done) is very similar to class 1 (punished). Fact was simple, pilots randomly had their bad days, no one was perfect for everytime. When chiefs punished students, they saw an improvement in the next flight, which will statistically tend to happen in anyway. This study also showed Israeli officers that class 2 (encouraged) was a little more successful than other classes. So, they have changed their methodology for performance improvement.

Well, Israeli Air Force is not related to our daily lives, at least not directly. So let me give you another example. Now the question is: what is the probability of a company to return to an e-mail in which sender proposes a master thesis research topic?

First we need to define active variables.Some might say it depends on how well the e-mail is written. Another parameter can be the sending time or date. According to a random web page that I found on internet, weekday-afternoons are the best times for sending e-mails[3]. Choosing the right company (in this case “right company”implies a company which is related to your thesis topic field) is also important to get a response. Finally, I believe “hot hand” concept can be applied to our answer to this question as well. If Kobe sends a lot of e-mails, his hands get warm and increases the probability of finding a company.

In this post, I wanted to hunt down the deceptions of probability and statistics. We have to struggle against damn lies. J

[1] Scientific American Magazine, News in Advances, p.14 February 2012 [2] Predictably Irrational: The Hidden Forces That Shape Our Decisions, Dan Ariely, Harper Perennial, 2010 [3] http://kb.mailchimp.com/article/when-is-the-best-time-to-send-emails/