Normal Distribution and the Lottery

Date: 20 Jun 1995 11:45:29 -0400
From: Mike Kelly
Subject: Random Raving

Dr. Math,

For a friend, I wrote a Visual Basic program to generate "Pick 6" 
lottery numbers, producing six random numbers from 1 to 46.

Curious, I re-wrote the routine to generate sequences until it hit a
pre-determined sequence, say 1, 6, 12, 29, 39, 44, or a stop limit 
of 100,000 iterations. When five of the six were matched, I kept 
track of which digit was not matched and found that the non-matches 
occurred on 12 or 29 more often than on 6 or 39 and much more often
than on 1 or 44, the result being a somewhat normal distribution, with 
the no-match count being, on one particular run, 6, 9, 15, 12, 9, 5, 
for the six pre-defined numbers respectively. This pattern was born 
out over numerous runs.

I have read something about random number generation being related 
to a normal distribution but can't understand why this is the case 
given the random nature of the routine. I'm not looking to win the 
lottery, but I'd appreciate your explanation of this phenomenon.

Mike Kelly

Date: 21 Jun 1995 15:14:30 -0400
From: Dr. Ken
Subject: Re: Random Raving

Hello there!

Well, I'm inclined to say that the short answer is "your numbers aren't
really random."  I don't know about the particular algorithm your program
uses (and I don't know all that much about _any_ random number 
algorithm), but I do know that if the numbers are truly random, they have 
the property that one number is roughly as likely to be chosen as any 
other number.

In your case, if your program is truly picking six independent random
numbers (that is, if the numbers that have already been chosen have no
effect on the choice of the remaining numbers) then your intuition is good;
there's no reason why you should get a normal distribution.

Think of it this way: in your program, there's no significance to the
concept of the order of numbers.  Instead of the order 1, 2, 3, 4, ..., the
numbers might as well be ordered 6, 39, 14, 1, 20, ... and there should be
no difference in the Lottery.  The Lottery doesn't care that 1 is lower than
2, which is lower than 3, and so on, it just cares that there are 40 (or
whatever it is) little symbols, and you can't have more than 1 of the same
symbol whenever you pick 6.  Thinking this way, there can be no legitimate
way to get a consistent imbalance of any kind.  So the problem must lie in
your random number generator, and I'm not sure what would be going 
on there.

Thanks for the question!

-K