Calculating Lottery Odds: How Are Your Chances of Winning Determined?

[Gomar]

I'd like to calculate the odds of winning the lottery. Each game has posted
odds, however, my calculations give different odds. Perhaps I am not doing it right, or
they use a different method, but here are my results compared to theirs:

If we use #'s 0-9, and pick one number, then odds of winning are 1 in 10. Thus, I just have to buy 10 tickets, play all numbers, and will get a winner 100%.
If 3 numbers are picked, then it's 10x10x10, or 000-999, thus 1:1000 correct?

But, in a single Take5 game, numbers cannot repeat. Key space is 39.
thus 39x38x37x36x35 = 69,090,840
To win top prize(all 5 #'s matched), their odds are 1:575,757.

http://www.nylottery.org/ny/nyStore/cgi-bin/ProdSubEV_Cat_405_SubCat_337562_NavRoot_300.htm

How is that calculated?

Pick10 has better odds, eventhough the key space is 80. Since 20 numbers are picked(1/4), and you pick 10(1/8), top prize matches 10, you have 2x chances to win. But, that's the same odds as matching 5 to 10 if key space was 40.

http://www.nylottery.org/ny/nyStore/cgi-bin/ProdSubEV_Cat_406_SubCat_337554_NavRoot_300.htm

Winning Numbers Matched Per Game: 10
Prize: $500,000
Chances of Winning
on one Game Panel
1:8,911,711

Isnt it 80^10?

In any case, how is that any better than playing slots in Vegas or A.C.? 1c, 5c, 25c, gives more games than $1 on above. I played 1c at a casino in Vegas, and won $10; that's 1000%.

 

[CRGreathouse]

But, in a single Take5 game, numbers cannot repeat. Key space is 39.
thus 39x38x37x36x35 = 69,090,840
To win top prize(all 5 #'s matched), their odds are 1:575,757.

You're assuming that order matters, that 12-32-11-15-02 is different from 32-12-11-15-02. The lottery commission assumes that order doesn't matter. (Redo your calculations with 5 numbers; you'll get 120 ways of choosing a ticket instead of 1.)

 

[statdad]

"You're assuming that order matters" - just to support CRGreathouse on this; it's easy to make this error (many of my students do so). A reason, one that they give as justification for their error, is "the winning numbers are always presented on tv/in print in increasing (or decreasing) order". That is true, but irrelevant to the game: it's done simply to make it easier for the audience to read/remember.

 

[Gomar]

Ok, so are you both saying that if the winning numbers are: 34-19-2-8-27 shown on TV as: 2-8-19-27-34, I still win even if my numbers on my ticket are printed as: 19-8-34-2-27?

I saw nowhere it says order matters or not, except for Daily Numbers, so perhaps I was wrong. That explains why several people win prizes instead of just 1.
However, it would seem logical that is 1 person did indeed get the order right, only he/she should get the prize.

 

[statdad]

"Ok, so are you both saying that if the winning numbers are: 34-19-2-8-27 shown on TV as: 2-8-19-27-34, I still win even if my numbers on my ticket are printed as: 19-8-34-2-27?"

Yes - the major lottery games, the ones I'm aware of, consider the winning set of numbers: as long as you have every one of the selected numbers, you win at least a share of the grand prize.

"However, it would seem logical that is 1 person did indeed get the order right, only he/she should get the prize."

If order of selection were specified as the requirement for winning, grand prize winners would occur so rarely that the game would (pure speculation on my part here) fall quickly from favor and nobody would purchase tickets.