Something strange happened in the Philippines on October 1, 2022: 433 people won the jackpot in the local lottery. For this particular lotto, six numbers with values ranging from 1 to 55 were randomly selected, and 433 winners were matched. Even stranger, when arranged in ascending order, the winning numbers were: 9, 18, 27, 36, 45 and 54. In other words, the winning numbers were multiples of 9 (9 × 1, 9 × 2, 9 × 3, and so on). strange coincidence attracted international attention And allegations of questionable behavior.
So, mathematically speaking, how likely is this outcome? When drawing numbers for a lottery, six numbers are chosen randomly out of 55 possibilities, with no repeating numbers. It does not matter in what order the numbers are drawn. We describe the number of possible combinations resulting from this process, such as 55 choose 6. Which is equal to approximately 29 million. So the chances of these exact numbers being drawn are about 1 in 29 million. But Every other possible outcome has an equal probability of about 1 in 29 million. To mathematically investigate the probability of fraud, you have to flatten the complexity and engage with Bayesian probability, which is what the renowned mathematician and Fields Medal recipient Terence Tao did. In the blog of October 2022 Post.
The real question we want answered is this: What is the probability that these numbers will be drawn in a rigged lottery? And, it turns out, it is quite difficult to determine, because it not only depends on mathematically calculable quantities, but also requires assumptions that do not necessarily have a scientific basis.
On supporting science journalism
If you enjoyed this article, consider supporting our award-winning journalism Subscribing By purchasing a subscription, you are helping ensure a future of impactful stories about the discoveries and ideas shaping our world today.
subjective statistics
If you are familiar with statistics, at this point, you may realize that we need to work with multiple hypotheses and the theory of Bayesian probability to examine this lottery scenario. If these topics are unknown to you, don’t worry! I can help.
First, there are essentially two hypotheses for us to consider. In the null hypothesis, we assume that nothing has been manipulated and that the lottery draw is completely fair. In the alternative hypothesis, the lottery was manipulated in some way. You can already see that this can be complicated because “somehow manipulated” is a very vague term and can include many scenarios. But let’s try to work with what we have.
The process in Bayesian statistics is as follows: First, determine the probability that which of the two hypotheses is generally true. In other words, do we believe that these lotteries are fair, or do we believe that they are more often rigged? Without much inside knowledge on the subject this question is subjective. Some may assume that most of the time the images are unbiased, meaning the null hypothesis is supported. Meanwhile, skeptics may believe that the alternative hypothesis is more likely.
Next, we can calculate how much an event changes these subjectively determined probabilities using Bayes’ theorem. The event in this case, of course, is that the six drawn lottery numbers are significantly multiples of 9. What is the probability that the event (drawing) occurs under the assumption of the null hypothesis? We calculated it at the beginning: The answer is about 1 in 29 million. And what is the probability that the event occurs under the assumption of the alternative hypothesis? Now it becomes difficult again because we have to think about scenarios in which the game could be rigged.
For example, the October 1, 2022 lottery could have been manipulated by corrupt officials who fixed the numbers before the draw to share the winning results with a select few. If they drew the numbers randomly, the probability of the above event occurring under the alternative hypothesis would still be 1 in 29 million – after all, in this scenario, corrupt individuals would have drawn the numbers fairly; He may have done this even before the official draw.
Because the probabilities of the event occurring under both the alternative and null hypotheses are equal, they cancel each other out. This means that the probabilities of the null hypothesis and alternative hypothesis remain completely unchanged from this dubious draw.
Of course, another assumption could be that if the corrupt officials wanted to manipulate a specific draw and randomly chose the numbers in advance, they would have rejected such specific sets of numbers as 9, 18, 27, 36, 45, and 54. With this in mind, the probability of the event occurring under the alternative hypothesis is reduced, making that scenario less likely.
a broken machine
Another possibility is that this was not a deliberate manipulation but the result of a faulty machine that did not randomize the numbers correctly. In this version of our alternative hypothesis, there are still more variables to consider which make probability difficult to determine.
For example, one might assume that a machine had malfunctioned, causing only unusual sets of numbers between 1 and 55 to be drawn with obvious patterns. If so, then the probability we are looking for is the probability using the set of all such unusual number groups. Under this assumption the probability of drawing six multiples of 9 is quite high, so it would appear that the alternative hypothesis is supported, but the hypothesis is almost rejected by the next lottery on October 3, 2022. That day, the numbers drawn were 8, 10, 12, 14, 26, 51 – a group of numbers with no apparent pattern. If this new phenomenon is included in Bayesian statistics, it reduces the probability of the alternative hypothesis to almost zero.
As Tao says, there are other alternative hypotheses we can explore, but these also fail to produce any concrete results. He identified three properties that an alternative hypothesis must meet to be statistically relevant:
-
The hypothesis should generally have a good chance of being true.
-
The alternative hypothesis must have a much higher probability of causing the specific event than the null hypothesis.
-
Given later observed events, such as subsequent lottery results, the alternative hypothesis should still make sense.
Those three points provide us with a guide for evaluating unusual events and questioning whether something might be wrong.
At the end of his blog post, Tao turned his statistical attention to another question: Why would so many people – all 433 – choose these same six numbers? Perhaps many people select numbers according to a specific pattern, such as a sequence of multiples of 9. A more convincing argument arises when one considers the layout of lottery tickets in the Philippines, which arrange the numbers 9, 18, 27, 36, 45, and 54 along a diagonal. In other words, that geometric pattern could explain how people chose these numbers.
This article originally appeared in spectrum der wissenschaft And was reproduced with permission.
