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
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
- Each of the numbers from 1 to 31 is chosen more
often than each of the numbers from 32 to 49. The simple reason for
this is that players like to use numbers corresponding to birthdays of
relatives or friends.
- Certain sequences of numbers are much more
commonly chosen than others. In particular, the sequence 1,2,3,4,5,6 is
normally chosen by some 7,000 players.
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:
- if the first sequence is chosen and those are
your numbers you will probably win about 714 pounds because you have to
share the jackpot with 7000 others.
- if the second sequence is chosen and those are
your numbers you will almost certainly win at least 1.6 million pounds
because it is extremely unlikely that more than 3 players in total
chose those particular numbers.
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:
- 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
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.
- 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/
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