How math can reveal lottery fraud
NEWS | 25 January 2026
On October 1, 2022, something strange happened in the Philippines: 433 people won the jackpot in the local lottery. For this particular lotto, six numbers ranging in value from 1 to 55 were randomly selected, and the 433 winners all matched. Even more bizarre, 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, etcetera). The curious coincidence drew international attention and accusations of suspicious behavior. So, mathematically speaking, how likely is this outcome? When drawing numbers for the lottery, six numbers are chosen randomly from 55 possibilities, with no repeating numbers. The order the numbers are drawn in does not matter. We describe the number of possible combinations this process can result in as 55 choose 6. Which equals roughly 29 million. So the probability of these exact numbers being drawn is 1 in roughly 29 million. But every other possible outcome has a similar likelihood of around 1 in 29 million. To mathematically investigate the possibility of fraud, you have to level up the complexity and engage with Bayesian probability, which is exactly what the renowned mathematician and Fields Medal recipient Terence Tao did in an October 2022 blog post. The real question we want answered is: What is the probability that these numbers will be drawn in a rigged lottery? And that, it turns out, is quite difficult to determine, because it depends not only on mathematically calculable quantities but also requires assumptions that do not necessarily have a scientific basis. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. Subjective Statistics If you’re familiar with statistics, at this point, you may realize we need to work with multiple hypotheses and the theory of Bayesian probability to examine this lottery scenario. If these topics are uncharted for 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 the lottery draw is completely fair. In the alternative hypothesis, the lottery was somehow manipulated. You can already see this could get complicated because “somehow manipulated” is a very vague term and could encompass many scenarios. But let’s try to work with what we have. In Bayesian statistics the procedure is as follows: First, determine the probability of which of the two hypotheses is generally true. In other words, do we assume that these lotteries are fair, or do we assume that they’re rigged more frequently? This question is subjective without a lot of inside knowledge on this topic. Some people may assume that most of the time the drawings are fair, meaning the null hypothesis is supported. Skeptics, meanwhile, may believe the alternative hypothesis is more probable. Next, we can calculate how much an event alters these subjectively determined probabilities using Bayes’s theorem. In this case the event is, of course, the six drawn lottery numbers remarkably all being multiples of 9. What is the probability that the event (the drawing) occurs under the assumption of the null hypothesis? We already calculated this at the beginning: the answer is approximately 1 in 29 million. And what is the probability that the event occurs under the assumption of the alternative hypothesis? Now it gets tricky again because we have to think through the scenarios in which the game could be rigged. For example, the lottery on October 1, 2022, could have been manipulated by corrupt officials who fixed the numbers before the draw in order to share the winning outcome with a select few. If they drew the numbers randomly, then the probability of the aforementioned event occurring under the alternative hypothesis would still be 1 in 29 million—after all, in this scenario, the corrupt individuals would have drawn the numbers fairly; they just would have done so before the official draw. Because the probabilities of the event occurring under both the alternative and the null hypotheses are the same, they cancel each other out. This means that the probabilities for the null hypothesis and the alternative hypothesis remain completely unchanged by this suspicious draw. Of course, another assumption might be that if corrupt officials wanted to manipulate a specific draw and chose the numbers randomly beforehand, they would have rejected such a conspicuous set of numbers as 9, 18, 27, 36, 45 and 54. Taking this into account, the probability of the event occurring under the alternative hypothesis decreases, 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 didn’t randomize the numbers correctly. In this version of our alternative hypothesis, there are still more variables to consider that make it harder to determine probability. One could, for example, assume that there was a machine malfunction that caused only unusual groups of numbers between 1 and 55 with a discernable pattern to be drawn. If so, the probability we are looking for is one using the set of all such unusual number groups. The probability of drawing the six multiples of 9 is quite high under this assumption, so it would seem as if the alternative hypothesis is supported, but the hypothesis is virtually refuted by the next lottery on October 3, 2022. On that day, the numbers drawn were 8, 10, 12, 14, 26, 51—a group of numbers without a discernible pattern. If this new event is incorporated into the Bayesian statistics, it reduces the likelihood of the alternative hypothesis to almost zero. As Tao notes, there are other alternative hypotheses we can explore, but these, too, fail to produce a convincing result. He identifies three properties that an alternative hypothesis must fulfill to be statistically relevant: The hypothesis needs to have a good likelihood of being true in general. The alternative hypothesis must have a much higher probability of producing the specific event than the null hypothesis. The alternative hypothesis must still make sense, given subsequent events that have been observed, such as the outcome of later lotteries. Those three points offer us a guide for evaluating unusual events and the question of whether there might be something fishy afoot. Toward the end of his blog post, Tao turned his statistical attention to one more question: Why would so many people—a full 433—select these same six numbers? Perhaps many people tend to choose 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 a lottery ticket in the Philippines, which arranges 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 Spektrum der Wissenschaft and was reproduced with permission.
Author: Daisy Yuhas. Manon Bischoff.
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