Dear All,
A colleague posed this question, and since I don't play
the lottery (being a statistician) I thought I'd pose it generally.
Apologies if it is well known.
For non-lottery players - You choose 6
numbers between 1 and 49 inclusive for one pound. If you match all 6
with those generated by a random number machine you win the jackpot.
Ignore payouts for less than 6 matches, and the bonus ball. Two
punters each spend 5 pounds per week. Punter 1 has a bag with 49
numbered balls in it, and after choosing 6, replaces them in the bag
for the next go. Punter 2 has the same bag, but does not replace the
balls before each new draw.
Who has the best chance of winning?.
One way of arguing goes thus:
The chance of winning on one go is about 1/14000000.
Thus Punter 1's chance of winning on 5 goes is about 5/14000000.
Punter 2 has a chance of 1/14000000 of winning on the first go. On the
second, if any of the winning numbers appeared in the first set, then
the chance of winning with the second and subsequent sets is zero.
Thus the chance of winning with the second set is less than
1/14000000. Similarly with the 3rd, 4th and 5th. Thus the overall
chance of winning is less than 5/14000000 and so Punter 1 has a better
chance.
However, another way of looking at it is that the above imposes an
artificial ordering on the numbers. The lottery has no interest in
how or in what order the sets were generated - all it sees are five
distinct sets of 6 numbers, each with a 1/14000000 chance of winning.
Since there is a small, but finite chance of Punter 1 drawing the
same set of 6 numbers more than once in the 5 draws, thus Punter 2 has
a (slightly) better chance. Of course in reality Punter 1 would not
enter the same set of numbers twice, so the two methods are equal.
I favour the latter but my colleague (being of little faith) would
like a more expert opinion.
Mike
Mike Campbell
Professor of Medical Statistics with an Interest in Primary Care
Institute of Primary Care
Community Sciences Building
Northern General Hospital
Sheffield S5 7AU
Tel 0114 271 5919
FAX 0114 242 2136
email [log in to unmask]
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