You should not be surprised when improbable events, miracles and other unusual cases occur, not even when the same sequence of numbers wins the lottery twice in a row
A set of mathematical laws that I call the "improbability principle" makes it clear to us why we should not be surprised by the occurrence of coincidences. In fact, we should expect various coincidences to occur. One of the pillars on which this principle rests is the "law of really large numbers". This law states that given a sufficiently large number of opportunities, we must expect each unique event to occur, regardless of the likelihood that it will occur on each of the opportunities alone. Sometimes we tend to think that the number of opportunities for the occurrence of an event is small, while in fact the number of opportunities is extremely large. This misconception leads us to greatly underestimate the probability of the occurrence of the event. In such cases, we tend to think that the probability that the event will occur is minimal, when in fact the chance that the event will occur is quite high, and it is possible that the event is almost certain.
How is it possible for a huge number of events to occur without people noticing that they are occurring? The "law of the combination of events", which is also one of the pillars of the improbability principle, can explain this phenomenon. The law states as follows: the number of possible combinations of interactions between different elements in a group increases exponentially (exponentially) with the increase in the number of members in the group. The "common birthday date" problem is a well-known example of this.
The common birthday date problem is this mathematical problem: how many people should be in the room so that the probability that the birthday of two of them falls on the same day will be higher than the probability that there is no pair of people whose birthday falls on the same day.
The answer is only 23 people. If there are 23 people in the room, the probability that a pair of them will have the same birthday date is greater than the probability that such a pair will not be found.
If you didn't know the common birthday date problem before, this answer probably surprised you. The number 23 sounds suspiciously small. The way you came to this conclusion may be this: the probability of a certain person in the room having the same birthday as mine is one in 365. The probability that one of the people present in the room will have a different birthday than mine is therefore 364/365. If there are n people in the room, and the probability that each of the other people, whose number is n-1, has a birthday on a different date than mine is 364/365, then the probability that everyone's birthday date is a different date than mine is 364/365 × 364/365 × 364 /365 ×364/365 … × 364/365, that is, a product of 364/365 by itself n-1 times. If n equals 23, the result is 0.94.
Since this is the probability that none of them have a birthday on my date, the probability that at least one of them has a birthday date like mine is 1-0.94 (this follows directly from the work that there are only two possibilities: either one of the people has a birthday on my date, or none of them has Birthday on my date, so the sum of the probabilities must be 1). But, 1-0.94 equals 0.06. This is a very low probability.
But this calculation is wrong because you were not asked to calculate this probability, that is, the probability that one of the people present in the room will have the same birthday date as yours. The question was what is the probability that one of the people present in the room has a birthday on the date of some other person present in the room, that is, a pair of people celebrating a birthday on the same date. This calculation does include the probability that someone in the room will have a birthday on your date, which is the probability we calculated above, but it also includes the probability that two other people in the room, or more, will have a birthday on the same date, even if it is different from yours.
This is the stage where the number of possible combinations comes into play. While there are only n-1 different people in the room whose birthday may be on the same date as yours, there are a total of n×(n-1)/2 different pairs of people in the room. The number of possible pairs increases rapidly as the number of people in the room (n) increases. When there are 23 people in the room, the number of possible pairs is 253, more than 10 times the number n-1=22. That is, when there are 23 people in the room, there are 253 different possible pairings, but only 22 of them include you.
Now we approach the calculation of the probability that there is not a single pair of people in the room with a common birthday date. The probability that two people have birthdays on different dates is 364/365. Therefore, the probability that they both have birthdays on different dates and, in addition, there is a third person in the room whose birthday is different from theirs is 363/365 × 364/365. Similarly, the probability that all three birthdays are on different dates and, in addition, the birthday of another person, a fourth, also falls on a fourth date is 362/365 × 363/365 ×364/365. If we continue this way we find that the probability that there are not two people in the room celebrating their birthday on the same date is: 362/365 ×363/365 ×364/365 … ×343/365. This probability is equal to 0.49. Therefore, the probability that several of them have the same birthday is 1-0.49, or in other words, 0.51, a number greater than half.
win the lottery
We will now turn to another example of events whose probability of occurrence seems low even though it is high: lottery draws. On September 6, 2009, the numbers 4, 15, 23, 24, 35 and 42 were randomly drawn in the Bulgarian lottery. There is nothing surprising in this series of numbers. Although all the digits in these numbers are low, 1, 2, 3, 4 and 5, it is not that special. In addition, the series includes a pair of consecutive numbers, 23 and 24. But this happens much more frequently than is generally believed (for example, when people are asked to randomly pick six numbers between 1 and 49, they pick consecutive numbers less often than in a true random selection).
The surprising thing happened four days later. On September 10, the numbers 4, 15, 23, 24, 35 and 42 were drawn in the Bulgarian lottery, exactly the same numbers that were drawn a week earlier. The results of the lottery caused a kind of media storm at the time. "This is the first time in the 52 years of the lottery's existence that something like this has happened. We are shocked by this strange coincidence, but this is what really happened," the Bulgarian Lottery spokesman was quoted as saying in an article by the Reuters news agency on September 18. Bulgaria's Minister of Sports at the time, Svilan Nykov, ordered an investigation. Could it be a sophisticated scam? Were the numbers copied in some way?
In fact, this amazing coincidence is just another example of the improbability principle, or in other words, the operation of the law of large numbers is really enhanced by the law of combinations. First of all, many lotteries are held around the world. Second, the lotteries are held time and time again continuously for many years. Therefore, overall there are many opportunities for the lottery numbers to repeat themselves. Third, the combination law comes into effect. Whenever any series of numbers is drawn in a lottery draw it may be the same as another series of numbers that has already been drawn in any previous lottery draw. In general, if n lottery draws have occurred, there are n×(n-1)/2 pairs of lottery draws that can contain the same series of numbers.
The Bulgarian Lotto draw repeated in 2009 is a draw of six numbers out of 49. Therefore, the probability of any set of numbers winning is one in 13,983,816. That is, the chance that a certain pair of lottery draws will result in a series of numbers identical to that of any other lottery draw is one in 13,983,816. But what about the chance that any two of the three lotto draws will result in the same set of numbers? Or among 50 lottery draws?
All three draws have three pairs of numbers, but in 50 different draws there are 1,225 possible pairs. The Law of Combinations therefore comes into play. If we continue, we find that in 1,000 different lotteries there are 499,500 different pairs. In other words, if we multiply the number of lotteries 20 times, from 50 lotteries to 1,000 lotteries, the increase in the number of pairs of different lotto lotteries will be much sharper: it will multiply 408 times and rise from 1,225 to 499,500. We are entering the realm of the really big numbers.
How many lotto draws must occur for the probability of the same series of numbers coming up twice to be greater than 0.5? Meaning that the probability of this happening will be higher than the probability of it not happening? Using the same method we used to analyze the common birthday date problem, you can find that the answer is 4,404 lotteries.
If there are two lottery draws every week, that is 104 draws a year, it will take less than 43 years to reach that number. That is, after 43 years, the probability that any two lotteries will result in two identical series of numbers will be greater than the probability that the event will not occur. This puts a completely different light on the Bulgarian Lottery spokesman's statement that it was an extremely strange coincidence!
But this calculation only applies to lotteries in one country. If we take into account the number of lotteries all over the world, we find that we should actually be surprised if the same series of numbers are not occasionally drawn in different lotteries. Therefore, you will not be surprised to know that in the Israeli lottery draw held on October 16, 2010, the series of numbers, 13, 14, 26, 32, 33, and 36, were drawn, which were also drawn a few weeks earlier, on September 21. You will not be surprised by this, but after the lottery, applicants flooded the radio stations in Israel claiming that the results of the lottery were predetermined.
The case of the Bulgarian lottery was unusual in that the identical series were drawn in two consecutive draws. But the law of really large numbers, together with the fact that many lotteries are held around the world one after the other, show us that we should not be particularly surprised. Therefore, surely you will not be surprised that this has already happened before. For example, in the North Carolina lottery, Cash 5, the same set of numbers was drawn on July 9 and 11, 2007.
Another, somewhat frustrating, example of the way the law of combinations can cause different lottery draws to match is the case of Maureen Wilcox in 1980. She purchased tickets that contained the winning numbers for both the Massachusetts and Rhode Island lotteries. Unfortunately, the numbers on the Massachusetts ticket contained the winning numbers in the Rhode Island lottery, and vice versa. When a person buys 10 tickets for different lottery draws, he multiplies his chances of winning 10 times. But ten tickets mean 45 different pairs of tickets and draws. Therefore, the probability that one of the tickets will contain the winning series in one of the lotteries is more than 4 times higher than the probability that he will win the lottery. For obvious reasons this is not a recipe for winning a huge fortune. Matching a ticket of one lottery with the winning numbers of another lottery does not win you any reward, except for the feeling that the universe is mocking you.
The Law of Combinations applies in cases where a large number of objects or people interact with each other. Let's take for example a class of 30 students. They can interact with each other in many different ways. They can work as individuals, then there is only one option to divide them into 30 groups. They can work in pairs, and then there are 435 different possible pairs. They can work in threes, and then there are 4,060 different possibilities. And so on, until, of course, the option where they all work together, and then there is only one option to divide them.
In total, the number of possible groups that the 30 students can make is 1,073,741,823 - more than a billion. In general, if a group contains n members, it is possible to create 2n-1 subgroups from it. If n=100, the result is 2100-1, a number roughly equal to 1030, by all accounts a really big number.
But even if 1030 doesn't sound like a big enough number to you, consider the law's implications for the Internet, which today has more than 2.5 billion users, each of whom can in principle communicate with anyone else. This amounts to 3x1018 possible pairs, and 10750,000,000 possible groups. Even events with extremely low probability become almost certain if they are provided with such a large number of opportunities to occur.
So the next time you come across something that seems like an unusual and strange coincidence, remember the principle of improbability.
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in brief
Many times we tend to attribute a very low probability to events that are actually happening around us all the time. The mathematical law of really large numbers, and the law of combinations of events help explain this phenomenon.
It is enough that there will be 23 people in the room so that the probability that the birthday of two of them will fall on the same date will be 0.51. That is, greater than 50%.
In the Bulgarian Lotto, on September 6, 2009, the series of numbers 4, 15, 23, 24, 35 and 42 were randomly drawn. After four days, the exact same six numbers were drawn. In the North Carolina lottery, Cash 5, the same numbers were drawn on July 9 and 11, 2007. Weird? Not according to probability theory.
Copyright
Adapted from the book "The Improbability Principle: Why Coincidences, Miracles, and Rare Cases Happen Every Day", by David G. Hand, with the permission of the book publishers Scientific American/Ferrar, Strauss and Giro Ltd. (North America), Transworld ( United Kingdom), Ambo/Anthos (Netherlands), S.H. Beck (Germany), Compagnie des Letras (Brazil), Grupa Videoniceza Foxal (Poland), Locus (Taiwan), AST (Russia). All rights reserved David G. Hand © 2014.
About the author
David G. Hand is Emeritus Professor of Mathematics and Senior Research Fellow at Imperial College London. He previously served as president of the Royal Statistical Society. He is the author of Statistics, A Very Short Introduction (Oxford University Press 2008).
More on the subject
Duelling Idiots and Other Probability Puzzlers. Paul J. Nahin. Princeton University Press, 2000.
Symmetry and the Monster: One of the Greatest Quests of Mathematics. Mark Ronan. Oxford University Press, 2006.
The article was published with the permission of Scientific American Israel
3 תגובות
Fix:
"The probability of a certain person in the room having the same birthday as mine is one in 365" ??
Definately not !
The date of birth of other people (if they are not related to me in one way or another) is in no way affected by my date of birth.
All those present were indeed born within the 365 days of the year, but there is no necessity for them to be born in the same year or for the birth dates of the rest of those present to be different from each other and thus limit the number of options.
In fact, the probability that someone present was born on my date is: (1*365-365*365) * (all the years in which each person present was born).
So the minimum possible probability is: 1/(365*364) = 1 in 132,860.
And the maximum probability is 1 in 132,860*number of attendees.
(This probability presents a situation in which each of those present was born in a different year).
Who taught this professor math???
"The probability of a certain person in the room having the same birthday as mine is one in 365" ??
Definately not !
The date of birth of other people (if they are not related to me in one way or another) is in no way affected by my date of birth.
All those present were indeed born within the 365 days of the year, but there is no necessity for them to be born in the same year or for the birth dates of the rest of those present to be different from each other and thus limit the number of options.
In fact, the probability that none of those present were born on my date is: (1*365-365*365) * (all the years in which each of those present was born).
So the minimum possible probability is: 1/(365*364) = 1 in 132,860.
And the maximum probability is 1 in 132,860*number of attendees.
(This probability presents a situation in which each of those present was born in a different year).
Who taught this professor math???
To sum up the article in a short sentence, it can be said as follows:
Even a result that appears to be extremely rare may occur with a probability greater than 0.5
If the number of events and if the number of participants are large enough.
In this context, emphasis should be placed on the participants in order for the law of large numbers to hold
to be identityless.
If each participant has a unique self-identity then the law of large numbers will not hold.
For example: What is the chance that someone will win the lottery? Let's say about 2\1, 3\1, 4\1, after all there are millions of participants.
On the other hand: what is the chance that Mr. A. Ben Ner will win the lottery? About 14.000.000 because A. Ben Ner is a single participant. And as I know A. Ben Ner and his luck, his chances are even smaller.