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Can you improve your chances of winning the lottery even if you can afford just one ticket?

The UK National lottery jackpot is won by selecting the exact 6 numbers (out of the numbers 1 to 49) that are randomly selected by a machine.  Just as we showed for sequences of coin tosses it can be shown that any particular set of 6 numbers is equally likely to be drawn as any other (it can be shown that each has a probability of about 1 in 13 million). So your choice of numbers cannot improve your chances of matching all six numbers drawn. The set of numbers:

1 2 3 4 5 6

is just as likely to be drawn as, say:

5, 19, 32, 37, 40, 43

However, you can certainly improve your chances of winning a bigger jackpot prize if you choose particular numbers. In fact, incredibly, if you choose the second of the above sequences and these numbers all came up then you would win much more money than if you chose the first sequence and all the numbers came up. And this assumes the size of the jackpot is the same in both cases.

How can this possibly be?

It is because the ‘jackpot prize’ is shared between all players who have the 6 matching numbers, and players do NOT choose their 6 numbers randomly. In particular:

What this means is that if the winning six numbers include several numbers over 32 then there are almost certain to be fewer winning players than if the numbers are all less than 32. In fact, based on data from previous lotteries, and assuming a lottery jackpot of say 5 million pounds:

Although the sequence 1,2,3,4,5,6 has never been selected there was one closely related incident that confirms the above argument.

It occured just a few weeks after the UK National Lottery started. The maximum number of jackpot winners for any single draw up to that week was 4 (indeed assuming about 20 million tickets are bought, it is no surprise that there are typically between 0 and 4 winners -- click here to see why).  However, in this particular week there were no less than 127 winners of the jackpot prize. Given the massive publicity that jackpot winners had received up to that point, and given that all jackpot winners had won over a million pounds (or very close to it) there was much shock and even outrage when the jackpot winners discovered that they had won less than 8,000 pounds each. So how did this happen?  

It turns out that leading up to this particular draw one of the TV teletext services had a  'psychic' predict the lottery numbers. The psychic actually predicted three correct numbers.  Many thousands of players that week used those numbers specifically as a result of seeing the teletext article. And this had a 'double whammy' effect on the jackpot winners:

  1. Most of the players who used the psychic's numbers used all six and so got exactly three correct numbers. The Lottery rules are such that EVERY player who gets exactly three numbers correct is guaranteed a ten pound prize. The other prizes (for 4, 5 and 6 correct numbers) are calculated AFTER the  money paid to the ten-pound prize winners is deducted from the money in the pool that week. Because of the 'psychic' prediction a disproportionately high number of players won the ten pound prize and hence this significantly diminished the prize money left for jackpot winners.
  2. Many of the jackpot prize winners also deliberately used the 'psychic' numbers. But instead of using all six they had three of their own 'lucky' or 'preferred' numbers (e.g. corresponding to the birthdays of three loved ones) and additionaly used the three successful 'psychic' numbers. The probability of selecting 3 correct numbers from 46 is much higher than the probability of selecting 6 correct numbers from 49 (one in 15,000 compared to one in 13 million). So given that a large number of people used exactly the three correct pychic numbers it was inevitable that there would be a much higher number of jackpot winners than normal.  

It is also interesting to note that the rule on the ten pound prize winners is such that it is conceivable for the lottery to be bankrupted as a result of too many players in a particular week getting exactly three numbers correct. And in such a scenario it would also be certain that players who got 4, 5, or 6 numbers correct would receive nothing at all.

How might this happen? Suppose a popular and widely respected televison entertainer went on prime time TV and announced that he knew that the next lottery draw was being fixed and that the numbers were, say  5, 19, 32, 37, 40, 43. (Obviously this would have to be the first and only time he made such an announcement).  It is conceivable that in such a situation 200 million tickets would be sold with exactly those 6 numbers (say 20 million people buy an average of 10 tickets each). Now suppose that exactly three of the chosen numbers come up. Then, by law, the National Lottery would have to pay out 2 billion pounds in ten pound prizes. Even assuming that 300 million other tickets were sold that week the income from ticket sales is just 500 million of which 50% must go to 'good' causes. Even ignoring adminstration costs, the lottery would be faced with a loss of 1.75 billion pounds as well as the fury of many 'winners' (including jackpot winners) who will not receive a penny.  The lottery could never recover from such a scenario.

Also see this example for more surprising probabilistic information about the National Lottery.

Real-time statistics about the UK national Lottery can be found here: http://www.lottery.co.uk/stats/

Norman Fenton

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