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-   -   Lottery pick-3 odds? (https://www.mersenneforum.org/showthread.php?t=22248)

 schickel 2017-05-01 02:37

Lottery pick-3 odds?

So, I know just enough probability to get myself in trouble. For example: pick-3 games (pick 3 digits from 0-9 inclusive) have 1 in 1000 odds of winning on a [I]straight[/I] bet, ie. I play 395 and the computer picking that number should happen every 1000 draws or so.

(I like to think) I'm not naive enough to think that if a number hasn't hit in 900 draws that it is more likely to hit than any other number (assuming a fair random number generator and unbiased code[SUP]*[/SUP]), but I don't know how to work out what the odds are that if I were to pick a number, say 734, and play it constantly that I will have to wait more than 1000 draws for it to hit.

[SUP]*[/SUP] [SIZE="1"]California uses an "Automated Draw Machine" to pick winners for the Pick-3 and other games.[/SIZE]

 science_man_88 2017-05-01 02:47

[QUOTE=schickel;457985]So, I know just enough probability to get myself in trouble. For example: pick-3 games (pick 3 digits from 0-9 inclusive) have 1 in 1000 odds of winning on a [I]straight[/I] bet, ie. I play 395 and the computer picking that number should happen every 1000 draws or so.

(I like to think) I'm not naive enough to think that if a number hasn't hit in 900 draws that it is more likely to hit than any other number (assuming a fair random number generator and unbiased code[SUP]*[/SUP]), but I don't know how to work out what the odds are that if I were to pick a number, say 734, and play it constantly that I will have to wait more than 1000 draws for it to hit.

[SUP]*[/SUP] [SIZE="1"]California uses an "Automated Draw Machine" to pick winners for the Pick-3 and other games.[/SIZE][/QUOTE]

depends on if they can be redrawn etc. at last check if they can the odds of not picking n for y drawings is (999/1000)^y I think, which would put your odds of not picking it for 1000 at about 36.77% in theory.

 schickel 2017-05-01 03:31

[QUOTE=science_man_88;457986]depends on if they can be redrawn etc. at last check if they can the odds of not picking n for y drawings is (999/1000)^y I think, which would put your odds of not picking it for 1000 at about 36.77% in theory.[/QUOTE]If you mean redrawn within each draw, yes, it's 3 independent fields of 10 digits. In fact just within the last week, the winning number was 555 (I think.....let me check)

 schickel 2017-05-01 03:41

[QUOTE=schickel;457990]In fact just within the last week, the winning number was 555 (I think.....let me check)[/QUOTE]Sunday April 23 evening [URL="http://www.calottery.com/play/draw-games/daily-3/winning-numbers/?DrawDate=Apr 23, 2017&DrawNumber=14426"]draw[/URL]. Looks like it doesn't really pay to play triples.

 a1call 2017-05-01 04:17

At the risk of stating the obvious, the probability of picking any given 3 digits is
(1/10)^3, so from a theoretical point of view, given the large enough sample(of draws), you will get any given 3 digit combinations very close to 1/1000th of times. However studying historical lottery results you will find out that some numbers are biased and occur more often.
There are two different factors at work.
1. The draw is not truly random
2. There is observed influence in random draws, which [U]probably[/U] has a quantum basis. In quantum physics it is a basic principle that just observing an experiment will affect the outcome of that experiment. For example in the double slit experiment, observing which slit a photon passes through will collapse its wave function and destroy the interference fringes which would be otherwise present.

 schickel 2017-05-02 00:54

[QUOTE=a1call;457994]At the risk of stating the obvious, the probability of picking any given 3 digits is
(1/10)^3, so from a theoretical point of view, given the large enough sample(of draws), you will get any given 3 digit combinations very close to 1/1000th of times. However studying historical lottery results you will find out that some numbers are biased and occur more often.
There are two different factors at work.
1. The draw is not truly random
2. There is observed influence in random draws, which [U]probably[/U] has a quantum basis. In quantum physics it is a basic principle that just observing an experiment will affect the outcome of that experiment. For example in the double slit experiment, observing which slit a photon passes through will collapse its wave function and destroy the interference fringes which would be otherwise present.[/QUOTE]Yeah, maybe, but if it was really easy, someone would have exploited any bias by now. According to the FAQ, they have two different draw machines and two different ways to pick the winning numbers, with both being chosen before the draws are run. I imagine there would also be a test draw or two before the winning draw, so it would seem to be impossible to figure out any effect there.[QUOTE=science_man_88;457986]depends on if they can be redrawn etc. at last check if they can the odds of not picking n for y drawings is (999/1000)^y I think, which would put your odds of not picking it for 1000 at about 36.77% in theory.[/QUOTE]Wow, that seems high to me. That's really why I'm asking, since my instincts could be wrong.

So, if there's a ~37% chance of not drawing one number in 1000 draws, what would be the odds of missing a win in the following scenario: take 10 different numbers and play them for 100 draws.

Intuition would seem to say that 10(numbers)x100(draws) = 1000 "trials" which would seemingly make it a certainty that you would hit once, but I don't know enough about how to calculate the odds in that case.

 science_man_88 2017-05-02 01:06

[QUOTE=schickel;458073]Yeah, maybe, but if it was really easy, someone would have exploited any bias by now. According to the FAQ, they have two different draw machines and two different ways to pick the winning numbers, with both being chosen before the draws are run. I imagine there would also be a test draw or two before the winning draw, so it would seem to be impossible to figure out any effect there.Wow, that seems high to me. That's really why I'm asking, since my instincts could be wrong.

So, if there's a ~37% chance of not drawing one number in 1000 draws, what would be the odds of missing a win in the following scenario: take 10 different numbers and play them for 100 draws.

Intuition would seem to say that 10(numbers)x100(draws) = 1000 "trials" which would seemingly make it a certainty that you would hit once, but I don't know enough about how to calculate the odds in that case.[/QUOTE]

I think that would make it into (990/1000)^100 = 0.36603234127322950493061602657251738619 ( used PARI/GP both times I think)

probability would depend on so many things see:

[url]https://www.youtube.com/watch?v=lP58mP8Wchc[/url] standupmath video ( matt parker)

that may help you. it's also why the monty hall problem stumbles so many as the video they made on that shows.

 science_man_88 2017-05-02 13:27

they give example odds for one play at the bottom of [url]http://www.calottery.com/play/draw-games/daily-3/how-to-play[/url] but there's a lot to think about some numbers it doesn't matter which you play etc.

numbers with all the same digit only have one way of sorting them so any= exact and the two main things become the same ( down to the prize money or using the straight/box option to win both prizes). those that have 2 digits in common ( something like 27% of the numbers total if you don't include the overlap with the all three digits the same in the set) have only 3 ways to arrange themselves, whereas numbers with no digit in common have 6 ways to arrange themselves.

let x stand in for the same digits those with two or more digits the same include ( per digit x with some overlap e.g for x=6 xx6 x6x and 6xx are all the same number ( 666 the case where all three are the same digit technically))
x0x
x1x
x2x
x3x
x4x
x5x
x6x
x7x
x8x
x9x
xx0
xx1
xx2
xx3
xx4
xx5
xx6
xx7
xx8
xx9
0xx
1xx
2xx
3xx
4xx
5xx
6xx
7xx
8xx
9xx

if you take away the overlap you have 28 possibilities remaining ( including 666 111 etc only once not three times). edit:doh I now see that this is already explained in that link.

 chalsall 2017-05-02 20:23

[QUOTE=science_man_88;458122]if you take away the overlap you have 28 possibilities remaining ( including 666 111 etc only once not three times). edit:doh I now see that this is already explained in that link.[/QUOTE]

Just to share, you do understand that the house always wins (statically)?

There is a reason lotteries are not popular where gambling is legal.

 science_man_88 2017-05-02 20:57

[QUOTE=chalsall;458142]Just to share, you do understand that the house always wins (statically)?

There is a reason lotteries are not popular where gambling is legal.[/QUOTE]

my point was one of best case scenarios because some numbers get better odds in the setup than others do.

xxx will have the best win with straight/box combo
xxy xyx and yxx forms still have best win for them in straight/box combo but are at lower number of possibilities than the forms xyz, xzy, yzx, yxz, zxy, and zyx.

 chalsall 2017-05-02 21:10

[QUOTE=science_man_88;458143]my point was one of best case scenarios because some numbers get better odds in the setup than others do.[/QUOTE]

It's a bit like trying to teach calculus to a mouse.

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