but maybe they did last weekend
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Analysis: what we learned from last weekend's bumper Lotto draw and why you should pull 23 from your lottery numbers for a few weeks
As a statistician, I have only ever played the Irish lottery twice. The first time, I volunteered to pick the numbers for a syndicate at work. I picked 6 numbers randomly from the set 32 to 47, thinking that in the unlikely event that we won the top prize, we would be less likely to share it. Unfortunately, we didn't win.
The second time was last weekend. Like many other Irish people, I thought last Saturday's draw was a great giveaway, not to be missed. Even if nobody claimed the €19.1 million jackpot, which had remained unclaimed in a statistical anomaly for more than seven months, this would now be distributed among lower prize winners. In fact, basic logic dictated that a larger aggregate sum of money would be won in prizes than taken in ticket sales if the lottery sold fewer than 4.75 million tickets at €4 a ticket.
So how many players actually played on Saturday? Demand seemed unprecedented, with the National Lottery’s website crashing under the load. While the lottery doesn’t disclose their total monetary ticket sales for particular draws, they do disclose the total number of players who won each prize category. There was one lucky winner of the overall jackpot of €19.1 million, an astounding 149 winners of the match 5 + bonus prize (more on this later), 250 winners of the match 5 prize, and so on.
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From RTÉ Radio 1's Today With Claire Byrne, interview with Laura Scriney, the owner of Laura's XL Store in Castlebar, Co Mayo which sold the winning €19.1 million Lotto jackpot ticket
All in all, there were 357,389 prize winners, most of whom took away small sums of money, such as €9 or €3. The probability of winning any prize in the lottery from a single selection of 6 numbers is approximately 1 in 29 (or 3.5%), or 2 in 29 (or 7%) for a single ticket. In other words, for every 29 tickets bought, we would expect to find two lines that win some sort of prize.
Given there were 357,389 prize winners, this suggests roughly 357,389 x 29 / 2, or over 5 million tickets sold. While this is a rough estimate, more careful calculations suggest it is likely that the Lottery sold between 5 and 5.2 million tickets on Saturday. This range is based on the assumption that winning tickets (that is, tickets that win at least Match 2 + bonus) are independent: knowing that a particular ticket bought by Joe won a prize gives no information about whether a ticket bought by Jill also wins a prize. This assumption is not perfect and breaks down when a particular player buys lots of tickets, with partially overlapping choices of numbers.
Nevertheless, the lottery sold more tickets for the main draw than the total population of the country. They likely took in more than €20 million but probably less than €21 million in prize money from the main draw alone, and may well have made a profit on the draw had nobody won the jackpot. But there was in fact a single winner so the lottery had to pay out more than the €19.1 million they had promised to distribute among lower prize winners in the event of nobody winning the jackpot.
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From RTÉ Radio 1's Drivetime, mathematician Brien Nolan from DCU on the odds of winning the Lotto jackpot
Simple calculations from the lottery website show that the lottery distributed around €28 million in prizes from Saturday night’s main draw, or at least 133% of ticket sales, somewhat justifying many of us in our decision to play. To put this in context, regulations dictate that at least 50% of ticket sales must be distributed as prize money, but this is on average and much less is usually distributed when the jackpot is not won.
For instance, on Wednesday’s main draw, only around 22% of tickets sales seem to have been paid out in prize money. Using the same logic as before, we can estimate that the number of tickets sold on Wednesday, the draw immediately after Saturday, was likely between 310,000 and 330,000, a small percentage of the tickets (6 or 7%) that were sold in Saturday’s draw. This indicates the huge number of casual lottery players attracted by the historic nature of the draw who played on Saturday.
A peculiarity of Saturday night’s draw were the 149 winners of the Match 5 + bonus prize. In its own right, the Match 5 + bonus prize is extraordinarily unlikely to win, with odds of 1 in 1,789,596 (roughly one in two million). If players picked numbers completely randomly, one would need to examine around 133 million tickets (that is 1,789,596 x 149 / 2 tickets) to find 149 winners.
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From RTÉ's Your Politics podcast, Sandra Hurley and Micheál Lehane talk to Bernard Durkan TD about his National Lottery crusade
This doesn’t tally with the fact that at most 5.2 million tickets were actually sold and something a little odd seems to have happened. Another way of looking at this was that 37% (almost two in five) of the 399 lucky players who won a match 5 or match 5 + bonus prize won a match 5 + bonus prize.
At the time the bonus ball was drawn, there were 41 balls left in the drum. These 399 players had a 1 in 41 (2.4%) chance of their unmatched number (their number which didn’t match any of the 6 numbers drawn) matching the bonus ball. However, 149 (37%) of these players had picked 23 (the bonus ball drawn), a 15-fold increase on the number of players we’d expect to pick 23 if they were picking their numbers randomly.
How likely would this be if players did pick numbers randomly? The answer is astoundingly unlikely and effectively impossible
How likely would this be if players did pick numbers randomly? The answer is astoundingly unlikely and effectively impossible. If we think of this probability as a 1 in N chance, think of N as being the number of atoms in the observable universe - and realise that you are still dramatically overestimating the probability.
Of course, players don’t pick numbers randomly, and it seems lots of people like the number 23. If another bonus ball rather than 23 had been drawn, it’s likely that far fewer players would have won the match 5+bonus prize, and perhaps would have each received larger payouts. Given that the only viable strategy of picking lottery balls that may increase your winnings is to pick combinations that few other players pick, this signals 23 as a ball to avoid for the next couple of years.
The views expressed here are those of the author and do not represent or reflect the views of RTÉ
