Statistics around the balance of even/odd and small/big numbers


I am sure that all of you realise that lotto numbers often come out with a balance of even/odd numbers (e.g. 2-4,3-3,4-2) about 75% of the time. The same thing for small/big numbers. I was wondering if there was a statistical/probabilty explanation to this. For any ball drawn, isn't the probabilty of it being even or odd the same? Is this tendency explained by the small probabilty increase that the next ball will be odd if previous balls drawn are even?

sully16's avatar - sharan

There is, but I don't know how to explain it, there are people here who can so be patient.

Todd's avatar - Cylon 200.jpg

<Moved to Mathematics forum>

Please post in the appropriate forum ... thank you.

time*treat's avatar - radar
In response to stef

Not all lotto games are the same. The even/odd balance in lotto games depend on the pool size of the individual game.

The stats are explained by the total (odd + even) and individual quantities of even and odd numbers possible.

In a game where the draw pool is an odd number (e.g. 49) the trend will skew towards more odd numbers being drawn -- there are 25 odds but only 24 evens. When the pool is even (e.g. 48 or 52) the trend will be more balanced (24/24, 26/26)

In response to stef

If your tracking shows you that the balance of even/odd numbers is 75%, why would you care what the mathematicians say it should be? 

 I am not trying to be obnoxious here tho I fear that is the way this question will be read.  What I am getting to is the fact that you have done some research and found an interesting bit of knowledge.  Now put that knowledge to good use and wager accordingly.

What other bits of interesting knowledge have you gleaned from your research?

In response to stef


My 2 cents worth.

In any matrix the difference between the number of odd or even balls will be one or less.  Consider the

stats below for a 5-39 game which show the total for odd number sets within the game.

5-39 = 20 odd and 19 even

(% rounded)

overall sets 575757                last 750 drawings

0-odd = 11628       2%                 11    1%

1-odd = 77520     13%                 86    11%

2-odd = 184110   32%               250    33% 

3-odd = 194940   34%               257    34%

4-odd = 92055     16%               113    15%

5-odd = 15504       3%                 33    4%

You can see from the stats above that the results follow the universe of sets which show a small

bias for the 20 odd numbers over the 19 even. 

For any fixed filter such as odd/even  the predicted outcome can be calculated prior to the first set

being drawn.  This relationship is built within the matrix and in a fair game the results will always

follow the universe of possibilities.  If 34% of sets have 3 odd numbers then 34% of the draws

will also have 3 odd numbers.  Small deviations are expected from time to time but overall they

will follow this rule very close. 


Coin Toss's avatar - shape barbed.jpg
In response to stef


Do you have any results of any game to prove this out?

As RL said, over the long run any preponderence is going to prove itself. In a game with an odd number of total numbers, 1 being the start number, there will always be more odds.

In any game with any matrix, "balanced or biased" (one more number in one category, odd / even, high / low, iside / outside) over the long run the biased will show or it will come out 50/50.

There will be short term trends but that never cancels out the math as long as the game is on the up and up.

garyo1954's avatar - garyo
In response to stef

As everyone has stated, you have 20 odd and 19 even.  The odd has a slight advantage in this case. The confusion is the numbers.

Any NUMBER have equal chance since there is only one of each in the pool. However....THROW THE BALLS AWAY!

Instead let's use 20 Black Beans, those we'll call odd. And 19 White Beans, which we call even.

That changes our perception a little, eh?

We have a 20 in 39 chance of drawing a Black Bean (odd) on the first draw. Let's say we do.

We now have an equal chance of drawing Black or White on the second draw since there are 19 Black/odd and 19 White/even remaining in the container.

If the second draw produces another Black Bean (odd), the chances now favor the White Beans since there are 18 Black/odd and 19 White/even, remaining in the container.

The Law of Black and White Bean Expectations, maybe?

In response to garyo1954

Don Catlin addresses this indirectly very well in this article:

Here, he uses the example of the fact that in 5 ball lotto games there is a bell shaped distribution curve of the sums of all the five number sets.  He points out that if prizes were paid for matching your set's sum with the lottery draw's sum, it would be a good idea to choose sets whose sums appear the most often.

But this is not the game we play!

Winners hold the tickets with the most NUMBERS that MATCH, not their totals, whether they are odd or even, or how many unique digits are represented.

But don't take my word for it; Don makes it VERY CLEAR!

(You can buy his book at the Lottery Post Book store.)

garyo1954's avatar - garyo

Don Catlin did a good job of addressing something totally different. His article is his take on playing sum totals.

I see nothing in that article dealing with probablity of odd/even numbers or high/low numbers which is the discussion in this thread. Don is talking about drawing 5 white balls, numbered 1 to 53 from one hopper, and 1 red ball from a pool numbered 1 to 42 from another. He makes no attempt to examine the possibilities of odd/even or high/low balls being drawn.

Now, if you read the article carefully, for the first 5 paragraphs he tears down the idea that a Bell Curve can be a useful tool, leading one to a likely sum total. He states 'it's a scam,' 'a pitch,' 'and a scheme.'

But in the sixth paragraph he ends the article by saying,"Well, I'll go them one better. You tell me what lottery you have in mind and I will send you the most likely total for that lottery and I'll provide this information absolutely free of charge."

Oh really, Don? Where do you plan on getting 'the most likely total for that lottery?'

ca-dreamin*'s avatar - Lottery-065.jpg
In response to garyo1954

As long as the black and white beans have the numbers I pick it's all good : )

I tend to pick the odd numbers anyway.

In response to garyo1954

TO: garyo1954

CC: RJOh, Stack47

You have completely missed Don Catlin's point, and the reason I referenced hisarticle.  You apparently didn't notice that I used the word "indirect" whencomparing Odd/Even and Unique Digit Count observations, to Set Summations.  If youcan't see that these numerological patterns produce frequency distributions whichare analogous, I can't help you. Even if there were material differences among thedistributions, it would be of no consequence.

When people laugh at the idea of onlyusing 4 Powerball numbers when buying 3,168 tickets, it tells me they haven't noticedthat, collectively, over the year 2010, the 137 participants in MadDog's PowerballChallenge "purchased" 198,651 tickets containing the winning Powerball, compared to212,106, which is the number probability theory predicted!  (How could that happen RJ?)  Please read Catlin'sarticle again.

If the kinds of fallacious reasoning you use was restricted to lottery play, itwould be of little consequence because it does not cause you to lose any more moneythan the majority of people who select their numbers using random methods.  However,this is not the case.  People make errors of judgment in many responsible positions every day based on flawed logical conclusions similar to those that keep youbelieving that you have control over your chances of winning the lottery, when youdo not. For example, people who reason the way you do serve on juries and parole boards and havetoo often wrongly condemned innocent people because of their inability tocorrectly interpret evidence data.

The underlying causes of this kind of fallacious reasoning is not the result of alack of general intelligence.  To the contrary, even medical doctors in emergencyrooms have been documented making tragically wrong diagnoses under the influence oftheir erroneous belief that, for example, a patient with a certain condition was"DUE" in their triage because none had yet showed up that night, and they wereaccustomed to 6 or more by that time.

Based on your comments thus far, I don't expect any of this to effect your thinking.

What I hope is that others who happen by here will be moved to consider what I'msaying, and do more research.  Here is an article that approaches this from a psychological perspective. I hope you'll read it:

And for those new to this thread, here's Don Catlin's article:



       MadDog's Powerball Challenge

        01/02/2010  Thru  12/29/2010

            All Draws for 2010

             137 Participants

    Total Ticket Costs         $8,271,258
    Total Winnings             $1,772,590 

    Overall Gain/Loss         -$6,498,668
                         Expected           Actual
         Category       #Wins        ROI*     #Wins     ROI*
         0 WB + PB     133991       0.049    124791    0.045
         1 WB + PB      66990       0.032     64153    0.031
         2 WB + PB      10508       0.009      9275    0.008
         3 WB           23037       0.019     27240    0.023
         3 WB + PB        606       0.007       432    0.005
         4 WB             435       0.005       428    0.005
         4 WB + PB         11.438   0.014         0    0.000
         5 WB               1.610   0.039         4    0.097
                             -----              -----
         Total (Excl Jackpot)       0.174              0.214
         JACKPOT            0.042   0.325         0    0.000
       * ROI is The Amount Won Per Dollar Spent on Tickets.

garyo1954's avatar - garyo
In response to jimmy4164

TO: Jimmy4164

CC: RJOH, Stack47


RE: Quick Review of OP's Question

"For any ball drawn, isn't the probability of it being even or odd the same?"

"Is this tendency explained by the small probablility increase that the next ball will be odd if the previous balls drawn are even?"

Jimmy, these are the two questions posed for discussion in (this) stef's thread. You have missed that point.

Now you might convince someone you have proven a correlation between reality and adverse conditions affecting probablitity and the Law of Expectations, but I wager my last chocolate chip cookie, it is nothing new. You've merely discovered in your own way the reason it is called 'probability' and not 'actuality.'

I am saddened to hear about the glitch in your computer and internet connection that caused your mispost. I am certain we would have had a lucid and worthwhile discussion had you had the good fortune to have your enlightening views expressed in the proper thread.

With warm regards, knowing you shall correct the problem at the soonest,


In response to Coin Toss

About 80% of the 13,983,816 combinations in a 6/49 lotto game have 2 to 4 even numbers so over time about 80% drawing will have 2 to 4 even numbers. In 5/39 games, almost 66% (379,050 combinations) have 2 to 3 even numbers. It can be used as a filter but considering a full 18 number wheel with 8 even and 8 odd numbers has 18,564 combinations and a 2 to 4 even number filter only reduces the total by 13.12% and leaves us with 16,128 combinations.

As for a short term bias (0 to 1 even or odd numbers), unless you guess correctly when the bias will occur and have lots of money to spend on tickets, it's not a very good bet. (2.8 million combos have 0 to 1 even or odd numbers.)

In response to jimmy4164

"When people laugh at the idea of onlyusing 4 Powerball numbers when buying 3,168 tickets, it tells me they haven't noticedthat, collectively, over the year 2010, the 137 participants in MadDog's PowerballChallenge "purchased" 198,651 tickets containing the winning Powerball, compared to212,106, "

What is laughable, you still have no clue how people play lottery games. It's highly unlikely even 1 of the 137 participants played all 3168 combinations in one drawing and to suggest all 137 bought 3168 tickets four times a week for an entire year even for statistical analysis is height of stupidity. When are you going to figure out if people don't play that way, your statistics are worthless?

The rules of the game: 

WBs up to ( 12 ) from 1 to 59 

PB's up to ( 4 ) from 1 to 39

It says nothing about purchasing tickets or any fee for playing the game.

Congrats to:  ConstantlyB , Fja , Butch2030 , Maddog , Winlotta , and Raven62..

Congrats and thanks to all who played.

Does it mention where the winner is the player that won the most imaginary money or the lost the least in some drawings?

The fact is the winners are determined by the players matching at least 3 WBs or 1 PBs. I've seen very little mentioned of if any of the players actually play their numbers or how they play them if they do. You're either creating an imaginary scoring system or suggesting 137 players would wager $658,944 a year playing.

"If the kinds of fallacious reasoning you use was restricted to lottery play, itwould be of little consequence because it does not cause you to lose any more moneythan the majority of people who select their numbers using random methods."

It's called common sense that players know better then to spend $3168 four times a week using only 12 WBs and 4 PBs and if they don't know, they'll find out after losing their first bet. The challenge is to see who can pick the most numbers, not to see who wins the most or in some drawings loses the least imaginary money.

"The underlying causes of this kind of fallacious reasoning is not the result of alack of general intelligence."

It's fact your data is based on an imaginary wager of $8,271,258 so any thing you say about it should be prefaced with "Once upon a time". A while back there was a discussion about comparing purchased player picks with purchased lottery terminal quick picks and someone suggested any like number of randomly generated combos would be sufficient. That fallacious reasoning does result from a lack of general intelligence just like your fallacious reasoning that 137 players would spend $3168 four times a week for a year playing Maddog's Challenges or someone playing the same pick-3 in every drawing for 30 years.

You're obviously are very good at collecting and analyzing data; why not stick with real data based on actual play and maybe get a nice hit for yourself of help other players?


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