Luck (and/or chance) is often confused with intuition. After all, it’s normal that if an outcome seems unlikely, our first thought is that we’re dealing with an unusual and therefore highly “lucky” event. Take the lottery for example. The idea of ​​a one in fourteen million result occurring twice in four days may seem unlikely, to say the least. Yet it has happened more than once, and there’s an explanation.

The example of Bulgaria. In 2019, a comprehensive investigation is underway to uncover how it was possible for the Bulgarian lottery to give out the same winning numbers (4, 15, 23, 24, 35 and 42) for a week, and that these were the same numbers that were given out in the same competition four days earlier.

The truth is that the government launched an investigation to verify if there was any manipulation in the face of public outcry, but no irregularities were found. This was not a unique situation.

A replay of Canada. This story does not lead to Canadian authorities and the same paradox. It turned out when they decided to return the unclaimed prize money accumulated by millions of subscribers. What did they do? They bought 500 cars and generated 500 random numbers with a computer.

So whoever had that number would get the car. The surprise came when the authorities released the list and it turned out that there was a duplicate number. Yes, the winner got two cars and they were given to him. How was it possible that there was a duplicate number? This was not a unique situation.

German lottery. In Germany it was June 21, 1995. The resulting series of numbers was the same as the one that appeared on December 20, 1986. It was the first time in 3,016 draws that something similar had happened. What or who is playing with luck in this way?

The answer lies in a paradox that, as we will see, involves a lot of mathematics.

The birthday paradox. To understand this mathematical certainty given in all the previous examples, we must return to its simple formulation under the following question: How large must a group of people be so that the probability of two people having a common birthday is greater than 50%?

Of course, the first thing we intuitively think is that the odds are very low, but if we do the math, that odds jump to 50.7%, which is completely counterintuitive. The answer, and the magic number, is a group of only 23 people, which is surprising because it shows how likely it is that two people will share the day in a small group.

HE? Imagine you’re at a party of 23 people. Even though there are 365 days in a year, the odds of two people having the same birthday are not as low as you might think. Each new person in the group has several opportunities to meet others. For example, the first person has no one to match with, the second has one, the third has two, and so on. After all, there are so many possible combinations that the odds of matching skyrocket.

Numerically, with 23 people, there are 253 possible pairs who could share the same birthday. Although the chances of a particular pair being a match are small, the number of pairs makes the total probability about 50%. This seems paradoxical because our intuition underestimates how many opportunities there actually are for a coincidence.

Mathematical formula of paradox

How is it calculated? We want all of these people to have different birth dates. Let’s see: a given person has a given birth date. The probability that the second person does not match the first is 364/365. The probability that the third person does not match the previous two is 363/365, and so on up to 23 terms.

If we extend this calculation to 23 people, we get the probability that no two people share the same birthday (0.49), which makes the probability that at least one pair does so 0.51 (or a little over half).

In other words, the mathematical formula for the birthday paradox is used to calculate the probability P(n) of at least two people in a group. N people share the same birthday. In the formula, each fraction represents the probability that a new person does not share a birthday with the previous people, and multiplying these gives the probability that no one shares a birthday. Then subtract from 1 to find the probability that at least two people match.

I explain the Bulgarian case. Now that we have formulated the paradox, we will finish by describing the first lottery case in Bulgaria. This involved the random selection of six numbers from a pool of 49, and lottery officials said it was impossible to tamper with the machines. The draws were reportedly “conducted in the presence of a special committee and broadcast live on national television to prevent cheating.” This ensures that any set of six numbers will occur once in 13,983,816: the result of a combinatorial calculation expressed (and pronounced) as “49 chooses 6.”

Given the selection of September 6 (4, 15, 23, 24, 35 and 42), the chances of the same combination occurring again would be extremely small. But as with the birthday paradox, that is not the question we should be asking. The question might be: What is the probability that some/two of three draws will coincide? Or what is the probability that some/two of 50 draws will coincide?

Solution. As statistician David Hand, emeritus professor of mathematics and senior researcher at Imperial College London, explains, “After all, in three lottery draws there are three possible ways for two sets of numbers to match. In four draws there are six possible pairs; in five draws there are ten. By the time you get to 50 draws there are 1,225 possible pairs, and in 1,000 draws there are 499,500 possible ways for two sets of numbers to match.”

This greatly increases the odds, because with 4,404 drawings, the probability of two drawings coinciding exactly is higher, and as Hand points out, “if there were two drawings each week, which works out to 104 per year, that amount would take less than 43 years to complete the Drawings.”

Put this way, it’s not all that surprising that this happened once 15 years ago, or indeed that it has happened on so many other occasions. “Considering the number of lotteries worldwide, it would be surprising if draws were not repeated from time to time,” the researcher concludes.

Of course, the fact that this happened in a period of four days is even more unique.

Image | Santeri Viinamäki, Ian Barbour, public domain

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