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Related to this question, if I have 1500 or so jackpot results from a 6/49 lottery (numbers drawn, number of winners and prize per jackpot winner), how can I demonstrate that some numbers are less likely to be chosen by players than others? I don't have direct access to the distribution of numbers actually picked, of course.

Take, for example, the hypothesis that players are more likely than would be expected by chance to pick numbers corresponding to dates, ie from (1-31):

I find a positive correlation between the number of numbers > 31 and the prize-per-winner, and a negative correlation between the number of numbers < 32 and the prize-per-winner.

Also, the sum of the numbers is positively correlated with the prize-per-winner, and the number of numbers > 31 is positively correlated with a rollover event, when no-one wins.

What is the best way of approaching this with the data available?

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This is trickier than it looks!

First, do people pick dates as month/day or as month/day/year?

If the former, then if people pick dates (and only dates) then the 6 numbers would have to correspond to three dates. If they pick one date and 4 random numbers, then 2 of the numbers have to correspond to dates, and similarly for 1 dates. So, the number of numbers under 31 would be nonlinearly related to the prize.

If the latter, then either 2 or 4 of the numbers would have to be under 32, and again you'd have a nonlinear relationship.

Furthermore, not all numbers under 32 are equally likely to be in dates: only 1-12 can be in day numbers; 29,30,31 can't appear in February; 31 can't appear in September, April, June, November or February.

So, one thing you could do is a regression:

Prize ~ # of numbers 1-12 + # of numbers 13-28 + # numbers 29, 30 + # numbers 31.

You would expect negative parameters for all of these. I leave it to others to calculate the expected size of the parameters. If you are really into this, you could try a randomization test of actual dates.

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    $\begingroup$ I see what you mean about nonlinearity... I get correlation coefficients of -0.14 (numbers 1-12), -0.06 (numbers 13-28) and -0.05 (29,30). It makes sense to assume that all dates chosen will include the month, so I'll focus on the numbers 1-12, which do indeed seem to be over-represented, if I understand this (anti-)correlation correctly. $\endgroup$
    – Tom
    Feb 3, 2014 at 9:01

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