ON October 1, 2022, a total of 433 bettors won the 6/55 Grand Lotto jackpot prize worth P236,091,188.40. Based on the 9 p.m. draw of the Philippine Charity Sweepstakes Office, 433 people got the winning combination of 09-45-36-27-18-54. Each of them will take home P545,245.24.

So, what is the probability of having 433 winners in the 6/55 Grand Lotto? The way to answer this question is the same as computing for the probability of obtaining a perfect score in a 25-item, quadruple-choice quiz by the shotgun method (i.e., pure guessing). That is, one can employ the binomial distribution formula, which a casual Google search can yield.

The underlying assumptions of the binomial distribution are: 1) that there is only one successful outcome for each trial, 2) that each trial has the same probability of success, and 3) that each trial is mutually exclusive or independent of one another. In a 25-item, quadruple-choice quiz answered by the shotgun method, the probability of choosing the correct answer per item is 25 percemt, and the probability of choosing a wrong answer is 75 percent.

The probability of obtaining a perfect score (25 correct answers) is very small, but it is not impossible. The mean of the probability distribution is 6.25. This means that the most likely outcome is having six correct answers (18.28 percent chance). In fact, the probability of getting 11 correct answers at most (i.e., P(X≤11)) is 98.93 percent.

Suppose that some lazy students employ the shotgun method (e.g., shading all Bs) and get perfect scores because the equally lazy teacher happened to set each answer to B for ease of checking. What must be established, of course, is whether the answer key was leaked. If none was leaked, then the teacher should be sternly reminded to vary the answers to make the shading-all-Bs method much less likely to work. After all, does being extremely lucky necessarily make one a cheater?

In the case of the 6/55 Grand Lotto, because arrangement does not matter, there are 28,989,675 ways of picking 6 numbers out of 55. So, the probability of picking the winning combination is 1/28,989,675, and the probability of picking a losing combination is 28,989,674/28,989,675.

According to the 2020 Census of Population and Housing conducted by the Philippine Statistics Authority, as of May 1, 2020, the total population of the Philippines was 109,035,343. Using this figure as an estimate of the current population size of the country, suppose that each Filipino is given a chance to place a bet in the 6/55 Grand Lotto. In other words, let there be 109,035,343 mutually exclusive and independent trials. Here are the computed probabilities of selected values of the random variable X, where X is the number of winners, and P(X) is the probability of observing X:

When Microsoft Excel is used to compute for the probability of having 433 winners, an error (#NUM!) appears because the value is way too small to be displayed entirely. In other words, the outcome is highly improbable, but not impossible.

The mean of the probability distribution is 3.76. This means that the most likely outcome is having three winners (20.62 percent chance). The probability of having eight winners at most (i.e., P(X≤8)) is 98.49 percent.

There are those who would like the October 1 draw to be investigated because, by sheer probabilities alone, there might be a reason to suspect an anomaly, which implies deliberate manipulation or collusion. Even if people cast their bets in certain patterns, as long as the draw itself was random, should there be a reason to doubt the outcome? There are strict protocols to ensure transparency and unbiasedness (displaying all numbered balls before making the draw, weighing the numbered balls, livestreaming the draw for everyone in the world to witness, etc.). Was there anything noticeably odd about how the draw was conducted then?

As the scientific precept known as Occam’s razor goes, all things being equal (i.e., that the October 1 draw was conducted the same way all previous draws had been conducted, presumably in a transparent and unbiased manner, as long-established protocols would ensure), the simplest explanation (i.e., that the outcome was due to pure chance) tends to be the right one. There are two key considerations in the way the lottery is conducted. One, the likelihood of success is very small. Two, the outcome is randomly determined on draw night. How likely was it for something really funny to happen during the draw that would determine an outcome of which certain individuals had already been aware?

*Dr. Ser Percival K. Peña-Reyes is the Director of the Ateneo Center for Economic Research and Development.*