Surprising stories, ordinary maths
Every so often a news story reports that the same numbers were drawn twice, that a person won a second time, or that several winners came from one town or workplace. Such stories feel too unlikely to be chance. Yet the probabilities involved are usually much less remarkable than they seem, because we are not asking about one specific event. We are asking whether anything of that kind happened anywhere, across all players and all draws.
A specific event versus any event
The distinction between a specific coincidence and any coincidence is the heart of the matter. The chance that you personally win twice is extremely small. The chance that somebody, somewhere, among millions of players over many years, has a second win is far larger. When we hear the story afterwards, we judge it against the first kind of probability, but what happened is the second kind.
The same reasoning underlies the birthday problem, in which a group of only 23 people has a better than even chance that two share a birthday, and more generally the coincidence myths that the birthday coincidence instinct article describes. A surprising match is not unlikely when there are many pairs that could match.
Repeated results across many draws
Take the illustrative 6 from 49 game with 13,983,816 possible results. Suppose it is drawn twice a week for 20 years, which makes 2 × 52 × 20 = 2,080 draws. How likely is it that the exact same six numbers appear twice at some point?
The number of pairs of draws is 2,080 × 2,079 ÷ 2 = 2,162,160. Each pair has a 1 in 13,983,816 chance of matching, so the expected number of matching pairs is 2,162,160 ÷ 13,983,816, which is about 0.155. That corresponds to a chance of roughly 14 per cent that some exact repeat occurs at least once in those 20 years. (This uses the approximation that 1 minus the exponential of minus 0.155 gives about 0.14.)
So an exact repeat is not something we should expect every year, but with an appreciable chance of 1 in 7 or so over two decades, it is far from impossible. A headline claiming “the same numbers came up twice” describes something that, across a large number of games around the world, would sooner or later occur.
Partial repeats are routine
Repeats of some numbers are even more common. In one draw of 6 from 49, the chance that a particular number appears is 6 ÷ 49, about 12.2 per cent. Consider two consecutive draws. The chance that at least one number appears in both is 1 minus the chance that the second draw avoids all six numbers of the first. That avoiding chance is 6,096,454 ÷ 13,983,816, about 0.436, so the chance of at least one shared number is about 0.564, or 56 per cent.
In other words, more than half of consecutive draws in this illustrative game share at least one number. An observer who notices that “17 came up again” is seeing something that happens more often than not. Treating this as a sign that the machine has a pattern, or that a number is hot, is an example of the thinking described under the gambler’s fallacy and the law of averages. The machine has no memory.
Winners in the same place
Clusters of winners in a town, shop or workplace follow similar reasoning. If tickets are sold across a region with many shops and many buyers, winners will land in some places and not others, and by chance some places will land several. Random events are not spread evenly. They form lumps, which the article on streaks and clusters in random sequences explains. A map of random points looks clumpier than most people expect, and the clumps do not signal anything unusual about a shop. Shops that sell more tickets will, unsurprisingly, see more winners too.
Groups can add to this. A workplace syndicate winning is one event shared by several people, not several independent wins, though it can be reported as a cluster of winners.
Why we notice the clusters
Several habits make coincidences seem meaningful:
- We count hits and ignore misses. Thousands of unremarkable draws pass without comment.
- Stories are filtered. Only unusual events make the news, so the sample we see is already selected.
- We look for patterns. Humans are good at seeing structure, even where there is none.
A fair conclusion
A fair draw can still produce repeated numbers, repeat winners and local clusters, because with many chances, surprising events become expected ones. None of it changes the odds for a new ticket, which remain as long as before. Seeing a cluster is not a reason to feel that a draw is “due” or that a particular place is lucky. Spending on tickets should stay within a limit you can afford, and anyone worried about their gambling can contact a local support service.