## 6/55 Magic

I was contacted by the research team for Kapuso Mo, Jessica Soho for their upcoming segment this Sunday regarding the “unusual” results of the PCSO 6/55 Lotto last September 28. On the jackpot draw, a total of 433 bettors were selected to split the 236 million peso prize. Even more interesting was the winning combination, the so-called “Lucky 9”: $(9, 45, 36, 27, 18, 54)$.

When rare events like this happen, the public goes on a frenzy. Journalists start knocking on university doors to try and get a professor to provide their analysis on the matter. With the specific case of the PCSO Lotto, the question hanging over the grapevine is whether the lottery was rigged. Probes are already being called putting into question the integrity of the results. Being a fan of Jessica Soho (her style of reporting became my de facto gold standard for human interest throughout my career in campus journalism), I could not turn down their request for an interview.

What I’m writing down on this blog post is essentially a carbon copy of the same mathematics I gave Jessica Soho’s team. For the benefit of those who don’t watch television, but mostly for myself, just in case I botched my computations for the show (for the future of my career I sure hope not), so I’ll have something to point people to just to prove I’m not a complete idiot. Let’s break down whether this 6/55 magic really was “magic”, or just another improbable event.

First off, winning the lottery is difficult. That in itself is already a rare phenomenon. It turns out you really are more likely to get struck by lightning than to win the jackpot. Obviously this depends on the game you’re playing (and your tendency to be outdoors during lightning storms). The PCSO 6/55 lottery in question started in 2020 consists of betting 6 numbers out of a possible pool of 55. Hence: six out of 55. You win the jackpot when you successfully get all six numbers of a winning combination that’s randomly drawn thrice a week.

Finding the probability of any winning combination requires counting all the possible combinations for this lottery. First, we use a simple counting rule: consider the act of betting on six numbers as choosing which from 1 to 55 to place in six slots. For the first slot, you have 55 numbers to choose from. For the second, because you can’t bet on the same number over and over again, you only have 54 to choose from. For the third, 53. This goes on until you have only 50 numbers remaining for your sixth bet.

$55 \times 54 \times 53 \times 52 \times 51 \times 50$

But this is what’s known as a Permutation. Meaning this computation is only valid when the order is an important consideration. This is not so for the lottery: meaning a bet of $(9,18,27,36,45,54)$ is the same as a bet of $(27,45,36,9,18,54)$. In fact, the winning combination for the September 28 draw was $(9,45,36,27,18,54)$. But so long as you’ve got all the same numbers, regardless of order, then you’ve won the jackpot.

So we need to adjust the original estimate. Doing so requires dividing by the total number of re-arrangements we can produce of the six slots. This is $6!$, which leads to what we call a Combination.

${55 \choose 6} = \frac{55!}{(55-6)! \times 6!}$

It turns out that there are 28,989,675 total combinations for the 6/55 lottery. That means that the “Lucky 9” configuration $(9,18,27,36,45,54)$ in whichever order they appear has a probability of exactly 1 in 28,989,675. Meanwhile, reported data suggests that a person has a 1 in 1,222,000 odds of being struck by lightning.

But this isn’t unique to the “Lucky 9” combination. Another less special combination, say, $(12,45,23,18,21,07)$ would have the same probability of 1 in 28,989,675. But this combination is just so mundane you’ll never see it on the news. So winning the lottery by itself is a rare event. If we go by the maxim that every improbable event is an evidence for some kind of anomaly, then surely we must investigate every single person who ever wins the jackpot?

We have a weird relationship with probability. This isn’t only true for the lay public, but even with graduate students and professors who regularly engage with statistics, as pointed out by Daniel Kahnemann and Aaron Tversky in their series of groundbreaking studies as summarized in Kahnemann’s popular book, Thinking Fast And Slow. The truth is, we don’t fully comprehend probability, and how they should influence our beliefs about random events.

Consider for instance the event of coming across a car with the license plate ABC 1234. If you’re in a quiet street, and you encounter this registration number, then it’s a rare event indeed. But the same is true for any registration number: the probability that you’ll encounter any specific registration number is a mindbogglingly rare event. But even if we consider this particular registration number ABC 1234, if you’re standing in a highway with hundreds or thousands of cars passing by everyday, then surely you will encounter it eventually.

The same is true for the PCSO lottery. The random draws is designed to give each of the almost 29 million configurations a probability of coming up. In 2022 alone there have been a total of 118 draws: sooner or later, the “Lucky 9” combination would come up. And it just so happens than when it did, it was a popular choice among bettors.

The second component of this problem is in the chances that you have 433 people winning the jackpot. After all, that one person wins is already rare enough. But more than 400 of them? People online have been calling bullshit. I say we take it easy. Put down the pitchforks just yet. I know our trust in our public institutions have been in a state of constant deterioration lately, but the lottery may just be the last bastion of fairness that exists in our country.

This second probability is trickier. To answer it, we need to determine the chances that people will bet on a “Lucky 9” combination. It should be noted that it’s difficult to compute the odds that people will bet on any particular combination. That is because betting attitudes aren’t exactly rational. People don’t choose numbers on their lottery tickets randomly, like you might when you’re taking a multiple choice exam and you absolutely did not prepare (but even here you may not be so cavalier, and try to make some educated guesses or find patterns in the professor’s design). People bet on lucky numbers. They bet on their spouse’s or their children’s birthdays. They bet based on what’s easiest to remember (various sources have pointed out that the “Lucky 9” configuration falls perfectly along a diagonal on the ticket).

Let’s assume for the sake of argument that people are betting randomly. Then we can calculate the probability based on the Poisson Distribution. For this we need an estimate of the average number of tickets that win the jackpot. Fields medalist Terry Tao, a god among mathematical men, has already worked out the math for this. For this let’s assume for the sake of argument that about one million bettors played for the draw. Taking Tao’s value of $\lambda = 0.03$, we find that the probability of 433 winners is a number so miniscule it’s mind-boggling.

$Pr(433) = \frac{\lambda^{433} \times e^{-\lambda}}{433!} \approx 10^{-1600}$

For the sake of comparison, quantum physics has a constant called the Planck’s Length, which represents the smallest length possible for any measurement in our universe. Anything smaller than this would be deemed too unstable to exist anywhere in our universe. That length is $1.62 \times 10^{-35} m$. Meaning the probability is so small, it can’t even be a valid physical measurement.

But again, people behaving rationally or randomly is more an exception than a rule. In actuality, people bet for the wildest reasons imaginable. Wang, et. al. (2016) have a study published in the journal Judgment and Decision Making that goes specifically into this. It turns out the “Lucky 9” combination is something of a favorite: it counts among the top 30 combinations that people bet on. In fact, out of 112,473 anonymous bettors whose data the researchers were able to collect, 350 played a similar combination 502 times.

Again taking for the sake of argument that 1,000,000 people played for the September 28 draw, even if just 0.03% played the Lucky 9 combination, then we’re expecting about 300 winners. Suddenly, the 433 winners don’t seem quite so special (other than the fact they just won half a million each, before taxes).

Terry Tao’s blog post goes into more rigorous detail on the mathematics, including a hypothesis testing framework based on Bayesian probability updating. Meanwhile, Jessica Soho even interviews some of the 433 winners of this stupefying event. I, for one, am happy that the public are once more engaging with statistics, but as always we need to be careful with how we use these tools.

Was there cheating involved in the 6/55 lottery draw? Maybe. But the numbers don’t give us any strong evidence to believe so. It’s a rare event in a universe that’s constantly laying witness to rare events (hell, life is a rare event, and yet we’re here). I think we should appreciate it for what it is. I suggest we put our attention elsewhere it really matters, like the confidential funds that our newly elected government has been so adamant about, and who’s footing the bill for the Marcoses’s F1 Grand Prix getaway. The magic, as always, is elsewhere.