Mathematics explains 433 sixes in the lottery in one day. Here’s how

October 1, 2022. Just like any other day in the Philippines. Except no. 433 people won the lottery. This is amazing when you consider that the winning combination was 6 numbers ranging from 1 to 55. These were 9, 18, 27, 36, 45 and 54. And this sequence itself is amazing because each number is a multiple of 9 (9×1, 9×2, 9×3, 9×4 and 9×5). This coincidence attracted many suspicions – and the situation was reported by media from all over the world.

Of course, mathematically such a distribution of numbers is possible, but… Exactly, what are the chances of this happening? The answer is – extremely rare. One in 29 million! But there is also a small twist – every other combination has the same chance.

To fully understand what happened here, you need to use a bit more advanced mathematics and Bayesian probability. And this is the method used by Terence Tao – a renowned mathematician and winner of the Fields Medal, the most prestigious award in this field. In short, Bayes says: you have a guess, and new information comes along – you change your assessment of that guess.

The whole world wanted to hear the answer to the question – what are the chances that such numbers will be drawn in a “fixed” lottery. Unfortunately, it is difficult to give such a clear answer because this is an area where mathematically calculable data clash with assumptions that are not necessarily based on hard scientific methods.

Generally, this is a topic where several different hypotheses must be used simultaneously to explain the problem – apart from the mentioned Bayesian probability.

To assess whether such a result could have been the result of chance or possible manipulation, at least two competing hypotheses must be considered. The first one – the so-called null hypothesis – assumes that the drawing was completely fair. The second – alternative hypothesis – suggests that some form of interference may have occurred. The problem is that “interference” can mean many things, from fixing the results to a failure of the drawing machine.

In Bayesian statistics, you start by determining which hypothesis is more likely overall – whether you assume that lotteries are usually fair or that they are more likely to be abused. This is always a subjective element. It then looks at how a new event—in this case, a set of six multiples of 9—changes these initial beliefs.

If we assume that the draw was fair, the probability of such an outcome is approximately 1 in 29 million. But under the manipulation hypothesis… it doesn’t have to be bigger. If someone “fixed” the results, but did so by selecting the numbers randomly beforehand, the chance of such a set is still 1 in 29 million. In such a scenario, a suspicious result does not increase or decrease belief in either hypothesis.

It can also be assumed that if someone actually manipulated the drawing, they would avoid such striking patterns as 9, 18, 27, 36, 45, 54. Then the probability of the alternative hypothesis actually decreases.

Another possibility is a failure of the drawing machine, which would generate numbers with unusual patterns. But this hypothesis quickly loses its meaning, because two days later, in the next draw, a set appeared in the next draw without any visible pattern. Once this new event is taken into account, the probability of failure drops to virtually zero.

Tao emphasizes that a reasonable alternative hypothesis must meet three conditions:
– it must be generally reliable,
– it must significantly increase the chance of the observed event,
– and must be consistent with subsequent draw results.

None of the analyzed possibilities meets these criteria.