Million-to-one lottery game

The odds of winning the jackpot in Mega Millions are almost 176 million to one. The odds of winning the jackpot in Lotto 6/49 are close to 14 million to one.

Everyone knows that a million-to-one chance pays off nine times out of ten. So specifying a reasonable field and a reasonable number of choices (i.e., you can’t have a million numbers in the field and you can choose only one number to win the jackpot), what would a lottery game in which the odds of winning the jackpot are exactly one million to one look like?
EDIT: Changed title from ‘One in a million’ to ‘Million-to-one’, in an effort to prevent confusion and pedantry.


Seven balls, each numbered 1-10. Must get an exact match on all seven balls. Odds are exactly one million to one.

I didn’t know that. I thought that a “million-to-one chance” pays off one time out of a million.

I suspect the OP is referring to fiction; any time a character in a book or movie refers to something as a “million to one” shot, it’s a lead-pipe cinch that it’ll happen.

[Removed link, which wanted to download.]

It’s a Discworld reference. Google “Million-to-one chance” pratchett, and it comes up as the first link for me.


That works with six balls, not with seven balls, if I’m understanding correctly.

One way to have a million-to-one lottery is to sell a million tickets, numbered from 1 to 1,000,000, then draw one number from the range 1 to 1,000,000. Having lotteries where the customers choose the numbers just confuses things, because it means that there is a chance that there is no winner, and there’s also a chance that two or more people are winners.

I think you mean six balls. Seven balls would give you 10[sup]7[/sup] = 10,000,000 possible results. Alternately, you could draw three balls from a set labelled 1 to 100. Or two balls from a set labelled 1 to 1000, but that’s starting to get a little silly.

Note also that in any of these cases, the balls must be replaced after selection (i.e., draw the first number, throw it back in the hopper, mix, draw the next number), and that order matters (1-2-3-4-5-6 is a different result from 2-1-3-4-5-6.)

Or have separate sets for each draw.

Ah, I see. (Obligatory TV Tropes reference)

:smack: Exactly.

Or, each digit is pulled from a separate pool of 10 balls. The Illinois Lottery does this with its “Pick 3” and “Pick 4” games, in which you’re picking a 3- or 4-digit number, and the numbers are generated with a separate ball machine for each digit.

But, yes, in any case, order would matter.

Or a million balls numbered from 1 to 1,000,000.:smiley:


So, it sounds to me like you want the solution to:

n choose k = 1 000 000

where k != 1.

Oops! missed that.

I think so. Unfortunately I don’t have the formula handy. (I just use the function on my calculator to enter the number of numbers in the field and the number of choices to be made from those numbers. e.g., [56] [enter] 5 [shfit-left + CnR].)

By ‘reasonable’ field in the OP, I mean something like a field of 30 or 40 numbers. Likewise the number of choices would be, say, four to six to be reasonable. (I don’t have my calculator at-hand, so those numbers are just made up to illustrate ‘reasonable’.)

The odds of winning the jackpot would be a million-to-one. The minimum prize would be $5, and would require three correct choices. Other prizes would be based on the number of choices matched, as is the case with other games. The minimum prize would always be $5. If there is no winner, the jackpot and other prizes would increase until there is a winner. The initial jackpot would be $500,000 to give the state the edge. Multiple winners would share the jackpot, as is currently the case.

If you’re constraining the question such that it’s limited to the standard lottery type games and rules (a group of balls, a fixed number of them chosen without replacement, order doesn’t matter), I don’t think you’ll find anything that meets your requirements. The best and closest to one in a million using your constraints is 44 choose 5: 1 in 1,086,008. 72 choose 4 gets us to 1 in 1,028,790 – closer, but well outside your definition of “reasonable field.” 25 choose 8 gets us to 1 in 1,081,575.


From the reference I found earlier:

:stuck_out_tongue: :frowning:

Unfortunately, you thought wrong. Odds of a million to one pay off one time in 1,000,001. In general, the chance of winning for odds of 1:n is 1/(n+1).

Of course, you’re right, though one in 1,000,000 is closer to the right probability than the “nine times out of ten” in the OP.