Benford's Law - Anticipation of coincidence and probability

Start | Overview of problems | Literatur | Credits 

Anticipation of coincidence and probability

If ordinary people were asked to estimate the probability of the number 1 to appear as the first digit of a randomly picked number in any newspaper, they would spontaneously say 1/9. This answer appears plausible because there are only 9 possible first digits, and there seems to be no reason to prefer a particular one.[1]As we have learned, this kind of logic actually contradicts the empiricism. The Bendford`s Law is not the only case where coincidence seems to contradict common sense. Psychologists have found many examples in their surveys about the psyche and coincidence, and have been looking for explanations of why people often have difficulties with the acquisition of statements about coincidence and probability.[2]

The following simple experiment may be done in class to start off a unit of stochastics to demonstrate the miscalculation of coincidence. The task is to throw a fair coin 200 times taking note of the sequence of heads and tailPage The students are allowed to choose their way of either throwing the coin 200 times or just making up a random sequence. The interesting part about this experiment is to see which students actually threw the coin and which made up the sequencePage In an actual sequence of coin-throws, you come across combinations of six or more repetitions (i.e. six times straight: heads) that are completely accidental. But in made-up-sequences these cumulations are avoided because they are not considered coincidental.[3]

You can also observe this in daily life. A basketball professional shooting three hoops in a row, is perceived as especially good in shape. These cumulations are mostly nothing but coincidental statistical cumulationPage [4] For our examples, this means that a cumulation of six times heads will occur more often as the coin is thrown more often. And there is nothing special about a professional basketball player who shoots many hoops, especially shooting three hoops in a row.

Another great example for wrong estimation of probabilities is the so called "birthday problem". Which is also known as "paradox of the first collision". It is about the probability of two people of a random group that have the same birthday.[5]The surprising thing about it is, that even with a
group of 23 people, the probability rises to more than 50% for two people to have a common birthday. The reason for this result, that first seems surprising, is that the quest is for the probability of any and not a certain "double birthday". Unconsciously many people think about the probability of any person to celebrate a birthday the same day as theirs birthday.[6]

According to psychologists it is hard for people to anticipate coincidences in the right way, because our brains are programmed to recognize relations between causes.[7]To find one`s way in the world, you need rules and no pat answer like "coincidence". So our brain works hard to argue coincidence away with different strategiePage One of them is selective perception. It only allows those information to come to our awareness that match our own expectations.[8]This trick lets people believe in telepathy or a deeper connection. For example, if a friend calls just that moment as one thinks of him, totally neglecting the fact of how many times one is thinking about a person and does not receive a call.

To draw a conclusion, you might say that in the area of coincidence and probability many things at first seem to contradict our intuition and common sense. To some degree there is a wrong perception of coincidence that should be discussed in a good stochastic class, so that the "birthday problem" for instance is understood by the students and not longer perceived a paradox. It should be the goal to make students understand how coincidence is something that does not seem to match our intuition at first, but abides by certain rulePage This is a necessary basic knowledge to interpret the probabilities predicted by Bedford`s Law.

Problem: 200 coin tosses (**)

If you have a chance, try to do the following experiment about 200 real and unreal throws of coins with an impartial candidate (i.e. sibling, student, parent, partner...). Only explain them in the end the meaning of the experiment. Please hand in your observations and the results gained from this experiment!

Problem: The birthday paradox (*)

Assess the probability of two people in a group of 23 persons having the same birthday!

Problem: Interpretation of statistic information (*)

The probability of a symptom-free person having a certain illness is 1%. A medical testing procedure is giving a positive test result of 99% of the persons concerned. A healthy person is still 2% likely to get a positive test result. How is a positive test result of a symptom-free person to be rated?

1. Try to answer that question according to your instinct! What would you say?
2. Assess the probability of a symptom-free person, who has been positively tested, to actually be ill!
3. What can be learned from a problem like this?

Problem: Percentage (*)

"This is great!" your son exclaims as you are shopping. "At first the castle was reduced by 20% and then an additional 30%. Now it costs only half the price."

What do you think about it?

[1] See Krämer (1990), Page 1.

[2] See Bach, Münter, Krüger, Rosenberg (2002), Page 3.

[3] See Hill (1998), S.364 and Hill (1999), Page 1-2.

[4] See Klein (2004), Page 109.

[5] See Henze (1999), Page 67-71 ad Krämer (2005), Page 28-33.

[6] See Krämer (2005), Page 29.

[7] See Bach, Münter, Krüger, Rosenberg (2002), Page 4.

[8] See Klein (2004), Page 108.