Lottery Senility

by Evil Nigel / 2008

The distributions of lotteries drawing different numbers of balls from
different pool sizes evolve at different rates. For a while I've been
searching for an objective method of comparing them. To this end I've
introduced a concept of 'lottery senility' based on the historical data 
available, for which my definition is:
A lottery is senile when the expected number of balls for each possible
frequency is less than 1.

For a simple example, take a 1 from 2 lottery.

Draw   Expected Ball Frequencies
No     0     1     2     3

0      2     0     0     0
1      1     1     0     0
2     0.5    1    0.5    0
3     0.25  0.75  0.75  0.25

So the lottery is senile after three draws.

I devised a simple spreadsheet macro to analyse various lottery formats
and calculate how many draws it would take them to become senile. For
comparison, I've also calculated the number of draws for the maximum
expected number of balls for each possible frequency to drop below the
square root of the pool size.

Lottery Format                         Senile  sqrt(pool)

5/50 (eg EuroMillions main draw):        4419     88
7/49 (eg UK lotto main plus bonus):      3120     62
6/49 (eg UK lotto main numbers):         3447     72
1/49 (eg UK lotto bonus):               19108    386
5/34 (eg UK Thunderball main draw):      1467     43
7/27 (eg UK Daily Play):                  604     22
1/14 (eg UK Thunderball):                 471     34
1/10 (eg each UK Dream Number):           177     17
2/9 (eg EuroMillions Star draw):           75      8
1/2 (trivial example):                      3      1


Lottologists using history data to analyse draws often have a preferred 
history length to work with. This information might be useful when 
comparing the amount of history available for alternative lottery formats.


(C) 2008 Evil Nigel
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