# The whims of a lottery

There has again been a lot of media attention for the results of a lottery which were wrongly believed to be fraudulent. Tuesday December 1st 2020, the winning numbers 5, 6, 7, 8, 9 and 10 were drawn for the South African Powerball lottery. The ‘normal’ lotto numbers 5, 6, 7, 8 and 9 were drawn from a range of 1 to 50 and the bonus number 10, the Powerball number, was drawn from a range of 1 to 20. In total, there were 20 jackpot winners. Each of them won 5.7 million South African Rand (around 310 thousand euros), while 79 participants had the correct normal lotto numbers, so 5, 6, 7, 8 and 9, but not the Powerball number 10. Each of these 79 participants won 6283 South African Rand (around 340 euros). The number of winners may seem large, but you have to account for the fact that a lot of participants do not choose their numbers randomly. They might use their date of birth, lucky numbers, mathematical sequences or geometrical patterns to choose their numbers. Filling in 1-2-3-4-5-6 as a sequence is the worst possible choice, but research has shown that this is in fact the most frequently filled-in combination. The probability that the six lotto balls fall into this particular sequence is of course the same as for any other sequence, but in the utterly improbable case that the six lotto balls fall into the sequence 1-2-3-4-5-6, the jackpot would have to be shared among many winners.

That lottery participants tend to follow the herd when choosing their lotto numbers is also nicely illustrated by the following. One of the most watched drama series ever on American television was Lost, which consists of six seasons that aired from 2004 to 2010. In the episode ‘Numbers’ of March 2nd 2005, the main character Hurley Reyes chose the six numbers 4, 8, 15, 16, 23 and 42 for a fictional drawing of the Mega Millions Lotto. With that combination, he won the fictional jackpot of 114 million dollars. In no time at all, these six numbers became the most popular sequence of lotto numbers aside from 1-2-3-4-5-6 and 7-14-21-28-35-42. This fact received a lot of publicity after the Mega Millions drawing of January 4th 2011 with a jackpot of 380 million dollars, a new all-time high. What happened then? From a range of 1 to 50, the six numbers 4, 8, 15, 25, 42 and 47 were drawn with 42 as the ‘Mega’ number, which was also chosen als Mega number in the tv series. The three smallest numbers (4-8-15) and the Mega number (42) matched up with Hurley’s choice in the fictive drawing in the tv series. In total, 41,763 participants had correctly forecasted these numbers to each win only 150 dollars.

The National Lotteries Commission of South Africa stated that the combination of six sequential numbers was unheard of and vowed to establish a committee to evaluate the lottery results.

Immediately after the South-African lottery draw on the 1st of December 2020, the validity draw was called into question on social media by many South Africans. ‘Lotto exposing themselves as a scam’, according to one Twitter user and another opined ‘Absolutely no way in hell that’s a coincidence’. Some thought a judicial inquiry would be the appropriate response, similar to the one that investigated the alleged corruption of ex-president Jacob Zuma. The National Lotteries Commission of South Africa stated that the combination of six sequential numbers was unheard of and vowed to establish a committee to evaluate the lottery results. Their spokesperson added that ‘If anything went wrong, we will report it’. It’s interesting to note that the result of the lottery draw in the South African Powerball lottery didn’t originate from a device with rotating balls in it, in presence of a notary, but from a random number generator on a computer.

The ‘exceptional’ is bound to occur sometimes, if you just wait long enough.

Undoubtedly the South African universities have skilled probabilists with ample knowledge of data analysis, and those will have certainly concluded that the controversy surrounding that lottery draw was a storm in a teacup: the ‘exceptional’ is bound to occur sometimes, if you just wait long enough. It is true that the probability of getting the numbers 5,6,7,8,9 and 10 in one single lottery result is extraordinarily small (1 in 42,375,200), but one has to consider that there are many lotteries in existence, and that each has an average of two lottery draws per week. Let us assume 200 lotteries existing worldwide, with each having two draws a week. For ease of calculation we assume that they are all 6/42 lotteries: every draw is comprised of 6 different random numbers between 1 and 42 (inclusive). The ‘exceptional’ occurrence under consideration is the event E, which is the event that the lottery result is a sequence of 6 consecutive numbers. This means that E can be any of the 37 possible combinations of 6 consecutive numbers between 1 and 42. The probability of occurrence of event E in a single draw in a single 6/42 lottery is 1 in 141,778. A simple calculation yields the result that during a one year period the probability of a 6/42 lottery having 6 consecutive numbers is equal to 13.6%. For a two year period this is 25.4%, and for a 5 year period 52.0%.

These calculations make the ‘impossible’ result seem, actually, quite possible. But alas, the media attention accorded to such reasoning is paltry compared to the outrage fueled headlines in the vein of “Is this 1 in 42 million lottery result a scam?”