# Million-to-one lottery game **URL:** \ **Category:** Factual Questions\ **Created:** [March 31, 2011, 4:04pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643 "2011-03-31T16:04:07Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [March 31, 2011, 4:04pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/1 "2011-03-31T16:04:07Z")
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. . ---
**Author:** ![kenobi\_65](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kenobi_65/32/13178_2.png) [@kenobi\_65](https://boards.straightdope.com/u/kenobi_65)\ **Post date:** [March 31, 2011, 4:12pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/2 "2011-03-31T16:12:49Z")
Seven balls, each numbered 1-10. Must get an exact match on all seven balls. Odds are exactly one million to one. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [March 31, 2011, 4:13pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/3 "2011-03-31T16:13:14Z")
> [@Johnny\_L.A](#): > > Everyone knows that a million-to-one chance pays off nine times out of ten. I didn’t know that. I thought that a “million-to-one chance” pays off one time out of a million. ---
**Author:** ![kenobi\_65](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kenobi_65/32/13178_2.png) [@kenobi\_65](https://boards.straightdope.com/u/kenobi_65)\ **Post date:** [March 31, 2011, 4:14pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/4 "2011-03-31T16:14:15Z")
> [@Giles](#): > > 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. ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [March 31, 2011, 4:15pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/5 "2011-03-31T16:15:09Z")
> [@Giles](#): > > I didn’t know that. I thought that a “million-to-one chance” pays off one time out of a million. [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. . ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [March 31, 2011, 4:17pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/6 "2011-03-31T16:17:57Z")
> [@kenobi\_65](#): > > Seven balls, each numbered 1-10. Must get an exact match on all seven balls. Odds are exactly one million to one. 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. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [March 31, 2011, 4:20pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/7 "2011-03-31T16:20:24Z")
> [@kenobi\_65](#): > > Seven balls, each numbered 1-10. Must get an exact match on all seven balls. Odds are exactly one million to one. 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.) ---
**Author:** ![Smeghead](https://avatars.discourse-cdn.com/v4/letter/s/f1d935/32.png) [@Smeghead](https://boards.straightdope.com/u/Smeghead)\ **Post date:** [March 31, 2011, 4:23pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/8 "2011-03-31T16:23:02Z")
Or have separate sets for each draw. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [March 31, 2011, 4:26pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/9 "2011-03-31T16:26:27Z")
> [@kenobi\_65](#): > > 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. > [@Johnny\_L.A](#): > > It’s a _Discworld_ reference. Google “Million-to-one chance” pratchett, and it comes up as the first link for me. Ah, I see. ([Obligatory TV Tropes reference](http://tvtropes.org/pmwiki/pmwiki.php/Main/MillionToOneChance)) ---
**Author:** ![kenobi\_65](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kenobi_65/32/13178_2.png) [@kenobi\_65](https://boards.straightdope.com/u/kenobi_65)\ **Post date:** [March 31, 2011, 4:39pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/10 "2011-03-31T16:39:21Z")
> [@Giles](#): > > That works with six balls, not with seven balls, if I’m understanding correctly. :smack: Exactly. ---
**Author:** ![kenobi\_65](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/kenobi_65/32/13178_2.png) [@kenobi\_65](https://boards.straightdope.com/u/kenobi_65)\ **Post date:** [March 31, 2011, 4:41pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/11 "2011-03-31T16:41:47Z")
> [@MikeS](#): > > 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, 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. ---
**Author:** ![Sparky812](https://avatars.discourse-cdn.com/v4/letter/s/eb9ed0/32.png) [@Sparky812](https://boards.straightdope.com/u/Sparky812)\ **Post date:** [March 31, 2011, 5:20pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/12 "2011-03-31T17:20:03Z")
Or a million balls numbered from 1 to 1,000,000.😃 ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [March 31, 2011, 5:23pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/13 "2011-03-31T17:23:45Z")
Ahem… > [@Johnny\_L.A](#): > > (i.e., you can’t have a million numbers in the field and you can choose only one number to win the jackpot). ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [March 31, 2011, 5:28pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/14 "2011-03-31T17:28:21Z")
So, it sounds to me like you want the solution to: n choose k = 1 000 000 where k != 1. ---
**Author:** ![Sparky812](https://avatars.discourse-cdn.com/v4/letter/s/eb9ed0/32.png) [@Sparky812](https://boards.straightdope.com/u/Sparky812)\ **Post date:** [March 31, 2011, 5:37pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/15 "2011-03-31T17:37:31Z")
> [@Johnny\_L.A](#): > > Ahem… Oops! missed that. ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [March 31, 2011, 5:49pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/16 "2011-03-31T17:49:52Z")
> [@pulykamell](#): > > So, it sounds to me like you want the solution to: > > n choose k = 1 000 000 > > where k != 1. 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. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [March 31, 2011, 6:02pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/17 "2011-03-31T18:02:33Z")
> [@Johnny\_L.A](#): > > 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’.) 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. ---
**Author:** ![Johnny\_L.A](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johnny_l.a/32/1084_2.png) [@Johnny\_L.A](https://boards.straightdope.com/u/Johnny_L.A)\ **Post date:** [March 31, 2011, 6:16pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/18 "2011-03-31T18:16:16Z")
Alas! From the reference I found earlier: > [@](#): > > It also has to be exactly a million to one - none of this fiddly “995,351 to 1” business, or whatever other number you might end up with. 😛☹ ---
**Author:** ![psychonaut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/psychonaut/32/4655_2.png) [@psychonaut](https://boards.straightdope.com/u/psychonaut)\ **Post date:** [March 31, 2011, 6:52pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/19 "2011-03-31T18:52:57Z")
> [@Giles](#): > > I didn’t know that. I thought that a “million-to-one chance” pays off one time out of a million. 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). ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [March 31, 2011, 7:12pm UTC](https://boards.straightdope.com/t/million-to-one-lottery-game/576643/20 "2011-03-31T19:12:22Z")
> [@psychonaut](#): > > 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. [Next page](https://boards.straightdope.com/t/million-to-one-lottery-game/576643.md?page=2)