SINGAPORE TOTO PRODUCES JACKPOT WINNERS BELIES PROBABILITY THEORY

Philippines has several lottery draws and the most difficult to win is the Grand Lotto 6/55. On September 2022, a strange event occurred. 433 people won a share in the jackpot by picking the correct 6 winning numbers. If that is not crazy enough, one punter bought 2 tickets with the same 6 numbers. The jackpot was pesos 236m, giving each winner about pesos 545,000. The probability of a punter picking 6 out of 55 numbers is 1 in 28,989,675. The probability of 433 independently picking the same 6 numbers is so small as to be impossible. Well, not quite. There could have been some sort of significant event that took place where gamblers conjure up some likely numbers, such as date of birth, death, age of some prominent person who died etc.. Nothing of the sort happened during that draw. Then again, there was something. The election was just over and Boongbong Marcos had won. Loyal supporters are appointed into various positions and loyalties are rewarded. Up and down the levels whether national or local, the same is played out. Pagcor, the regulatory agency taking care of gambling, has always been held as a cash cow for shenanigans.

So is Singapore's Toto lottery any cleaner? I recall decades ago, Toto agency outlets had a certain percentage reward for jackpot winning tickets sold at their counters. The outlets where winning punters bought the tickets used to be announced. This serves to authenticate the system since the winning outlets can vouch for it. For reasons unknown, this practice stopped many years ago. There's no way for the public to vouch winnings.

Singapore Toto is 6 out of 49. Picking the correct 6 numbers is simply random luck and a matter of probabilities. The number of combinations are 49! / 6! x (49-6)! This is factorial math and can be simplified to :  49x48x47x46x45x44 / 6x5x4x3x2x1 = 13,983,816. This means the 49 numbers can make 13,983,816 of different combinations of 6 numbers. One of the combination is the winning set. This means a punter buying a system 6 ticket has 1 in 13,983,816 chances of hitting the jackpot.

But it does not mean that there is a sure winner for every 13,893,816th ticket. It means that over a long period, the statistics will show on average there is a winner in every 13,983,816 thickets. Put another way, suppose each draw has a sales of 13,983,816 there is a good chance of a winner at every draw. But there could be a series of several draws with no winners.

I parsed the numbers for the 6 months from January to June 2024 to see what I can find. The number that I need to have is the tickets sold at each draw. Since 38% of sales collection goes into the jackpot and this figure is shown at Singapore Pools website, it is possible to determine the number of tickets sold. The total number of tickets sold during the 6 months were 333,004,263. There were 46 jackpot winners. This means the average winning was 1 in 7,239,221.

For the 6 months data, Singapore Pools' experience of a probability of 1 in 7,239,221 does not comport with statistical probability of 1 in 13,983,816. It is twice as easy to win at a Singapore Toto draw than statistically possible. Something does not add up. 

During the 6 months, there were 11 draws with 2 winners. The probability of having 2 punters to independently pick the same 6 numbers out of 49 is extremely impossible. The odds are (1/13,983,816) x (1/13,983,816) which works out to 1 in 195,547,109,921,856. The odds are so slim as to be in the reaml of impossibility and yet there were 11 such events out of 51 draws.

What about the chances of 3 punters independently picking the same 6 numbers? The probability is (1/13,983,816) x (1/13,983,816) x (1/13,983,816 and the answer to that is 1 in 2,741,171,473,428,326,816. That probability is actually saying it is impossible, and yet during the 6 months it happened not just once, but 3 times. 3 times out of 51 draws is preposterously testing credulity.

The probability of 2 or 3 punters drawing the same 6 numbers are infinitesmally small as to be impossible. However the randomness of statistics mean that by flukeshot, it can happen. But having 11 flukeshots of 2 winning punters, and 3 flukeshots of 3 winning punters, in just 51 draws, is a tale too tall. It's a smoking gun that something is not adding up.

The data for the draws in Jan - Jun 2024 is in the table below. There were no cascading draws, so less complications. Tickets are $1 for a system 6 set.

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