Here is another equation to determine the chance to have a number appear X times in the last Y draws of a lottery game (Pick 5/6/7). This uses the Bernoulli formula which I'll describe here. Again, assume we have a game a of b (eg 5 of 45, a=5, b=45) and we want to find out the chance to have a number to appear x times in the last y draws (obviously 0<=x<=y).

First, we have to determine the chance p to have a number appear in a draw. This is equivalent to find how many combinations contain our number among the whole possible combinations.

The combinations that contain a particular number are C(b-1,a-1) - the combination formula. In our example of a=5, b=45 we have exactly 135751 combinations from a total of C(45,5)=1221759 combinations. So, p=C(44,4)/C(45,5)=135751/1221759=0,111.

Now, we want to find out the chance P to have a particular number drawn x times in the last y draws.

The Bernoulli formula suggests that the result is

P=C(y,x) * (p^x) * [(1-p)^(y-x)] where k^l is the power of k to l.

An example: a=5, b=45 (game 5 of 45). What is the chance of a number to appear x=3 times in the last y=5 draws?

We've determined p=0.111 before.

So, the chance is C(y,x) * p^x * (1-p)^(y-x) =

= C(5,3) * 0.111^3 * 0.888^(5-3) =

= 10 * 0.00136 * 0.78854 = 0.0108 or 1.08% approx.

As you can see, this rate is rather low, so we can "safely" remove any numbers that have appeared 3 or more times in the last 5 draws.

Note that this rate is to have a number appear 3 times in the last 5 draws, which means that if we have a number appeared 2 times in the last 4 draws, then the chance to appear in the next draw is 1.08% (and therefore have the number appear 3 times in the last 5 draws).

The following tables are the results of the above equation.

Horizontal axis - X parameter

Vertical axis - Y parameter

Game type 5 of 45 (a=5, b=45)

p=0,111111111

Code: Select all

```
0 1 2 3 4 5
01 88,89% 11,11% ------ ------ ------ ------
02 79,01% 19,75% 01,23% ------ ------ ------
03 70,23% 26,34% 03,29% 00,14% ------ ------
04 62,43% 31,21% 05,85% 00,49% 00,02% ------
05 55,49% 34,68% 08,67% 01,08% 00,07% 00,00%
06 49,33% 37,00% 11,56% 01,93% 00,18% 00,01%
07 43,85% 38,37% 14,39% 03,00% 00,37% 00,03%
08 38,97% 38,97% 17,05% 04,26% 00,67% 00,07%
09 34,64% 38,97% 19,49% 05,68% 01,07% 00,13%
10 30,79% 38,49% 21,65% 07,22% 01,58% 00,24%
```

p=0,15

Code: Select all

```
0 1 2 3 4 5 6
01 85,00% 15,00% ------ ------ ------ ------ ------
02 72,25% 25,50% 02,25% ------ ------ ------ ------
03 61,41% 32,51% 05,74% 00,34% ------ ------ ------
04 52,20% 36,85% 09,75% 01,15% 00,05% ------ ------
05 44,37% 39,15% 13,82% 02,44% 00,22% 00,01% ------
06 37,71% 39,93% 17,62% 04,15% 00,55% 00,04% 00,00%
07 32,06% 39,60% 20,97% 06,17% 01,09% 00,12% 00,01%
08 27,25% 38,47% 23,76% 08,39% 01,85% 00,26% 00,02%
09 23,16% 36,79% 25,97% 10,69% 02,83% 00,50% 00,06%
10 19,69% 34,74% 27,59% 12,98% 04,01% 00,85% 00,12%
```

p=0,12244898

Code: Select all

```
0 1 2 3 4 5 6
01 87,76% 12,24% ------ ------ ------ ------ ------
02 77,01% 21,49% 01,50% ------ ------ ------ ------
03 67,58% 28,29% 03,95% 00,18% ------ ------ ------
04 59,30% 33,10% 06,93% 00,64% 00,02% ------ ------
05 52,04% 36,31% 10,13% 01,41% 00,10% 00,00% ------
06 45,67% 38,24% 13,34% 02,48% 00,26% 00,01% 00,00%
07 40,08% 39,15% 16,39% 03,81% 00,53% 00,04% 00,00%
08 35,17% 39,26% 19,17% 05,35% 00,93% 00,10% 00,01%
09 30,86% 38,76% 21,63% 07,04% 01,47% 00,21% 00,02%
10 27,08% 37,79% 23,73% 08,83% 02,16% 00,36% 00,04%
```

p=0,113207547

Code: Select all

```
0 1 2 3 4 5 6
01 88,68% 11,32% ------ ------ ------ ------ ------
02 78,64% 20,08% 01,28% ------ ------ ------ ------
03 69,74% 26,71% 03,41% 00,15% ------ ------ ------
04 61,84% 31,58% 06,05% 00,51% 00,02% ------ ------
05 54,84% 35,01% 08,94% 01,14% 00,07% 00,00% ------
06 48,63% 37,25% 11,89% 02,02% 00,19% 00,01% 00,00%
07 43,13% 38,54% 14,76% 03,14% 00,40% 00,03% 00,00%
08 38,25% 39,06% 17,45% 04,46% 00,71% 00,07% 00,00%
09 33,92% 38,97% 19,90% 05,93% 01,13% 00,14% 00,01%
10 30,08% 38,39% 22,06% 07,51% 01,68% 00,26% 00,03%
```

Lotto Architect,

Thanks for these tables they are a great resource and aid in determining numbers to play - or not to play.

I think I prefer this statistical approach rather than relying on some mystical, psychic or divine prediction.

draughtsman

Mr Lotto Architect,

The Bernoulli tables in this thread are a great resource in assisting one decide on which numbers might be avoided at the next draw where Bernoulli's probabilities are based on occurrences so far and the chance of a nunber occurring at the next draw if it has already had some recent occurrences. However I am wondering if these Bernoulli tables might be able to be prepared in the reverse so that they might express a probability for a number to occurr at the next draw based on that numbers non occurrence over the past few draws. Given that I quite confidently reject a number if its chance of occurrence at the next draw is at 1% or less I presume I could choose to include a number if Bernoulli suggestsed its chance of occurence was up near or in excess of 99%. Would such a table be possible please? Or perhaps I should ask first can such a table be prepared?

regards

relowe

The tables already describe this. You have simply to observe the 1 column. To identify the chance of a number not to appear in the next draw (given it has not appeared yet), read the value at column 0. To identify the chance the number to appear at the next draw, given it hasn't appeared in the previous draws, read the value in column 1.

What might look stange is that the rates are in decreasing order after a certain amount of draws (lower bound for 0/1 columns) because statistically a number owes to appear a "certain" amount of times in a given range of draws and this is reflected in the Bernoulli tables. Thus, we cannot really expect to have a number not to appear at all in the last eg 30 draws and thus the chance for this not to appear again in the next draw is very low (the number normally should have appeared already sometimes at previous draws). But, as luck controls everything, we cannot exclude that too. This is why we'll never have an 100% or 0% for an event to occur.

In the above tables, the most dominant areas in a 5/45 game for a number to occur (given it hasn't appeared in previous draws) is between 6 & 10 draws later (rates around 37-38% in column 1).

Perhaps you would expect a table that says eg if we haven't a success till draw eg 30, then the draw 31 should have a huge % to occur (as it has to occur sometime of course!). I'll have a look at this property and if I come out with something useful, I'll post it. There is no distribution that describes what you ask for. Even the geometric distribution, shows the chance to have a number occur after n draws of non-hit. This follows the Bernoulli tables above as well.

Mr LA Man,

Thanks for your time on this - so easy when you know how Those Bernoulli tables now take on even more meaning. And certainly appreciated if you find any other statistical clues on numbers to come at the next draw.

relowe

Hi Mr LA,

Could I trouble you please for an extra Bernoulli table - for a game type of 6 from 45.

Thanks

relowe

relowe, you can use the formula above to find any particular game you want. Even better, you can use Excel as a template to generate quickly Bernoulli tables for any type of game you want. If I have some time I'll provide those tables you ask but unfortunately at the moment I lack free time.

Mr LA.

Thanks for the challenge. Have included below a Bernoulli table for a 6 from 45 type lottery; I believe it to be correct. Perhaps any other more competent mathematician than I (and Excel exponent) might care to verify this.

Bernoulli Table

Lottery Type 6 of 45 (a=6, b=45) where p = 0.133333

Chance of a ball appearing X times (horizontal) in Y draws (vertical).

Code: Select all

```
......0......1......2......3......4......5......6..
01 86.67% 13.33% ------ ------ ------ ------ ------
02 75.11% 23.11% 01.78% ------ ------ ------ ------
03 65.10% 30.04% 04.62% 00.24% ------ ------ ------
04 56.42% 34.72% 08.01% 00.82% 00.03% ------ ------
05 48.89% 37.61% 11.57% 01.78% 00.14% 00.00% ------
06 42.38% 39.12% 15.04% 03.09% 00.36% 00.02% 00.00%
07 36.73% 39.55% 18.25% 04.68% 00.72% 00.07% 00.00%
08 31.83% 39.17% 21.09% 06.49% 01.25% 00.15% 00.01%
09 27.58% 38.19% 23.50% 08.44% 01.95% 00.30% 00.03%
10 23.91% 36.78% 25.46% 10.45% 02.81% 00.52% 00.07%
```

3 draws what is the chance of it being drawn 3 times in 4 draws (for example).

Or for any other draw occurrence and draw history ie if 4 times in 8 draws what

is the chance of 5 times in 9 draws? My choice is to avoid numbers where the chance is about 1% or less. Others may consider avoiding numbers where the chance of appearance may be say upto 5%. These tables have been quite accurate for me although I did have a recent occurrence where 1 number appeared 5 times in 7 draws. I think even Mr Bernoulli would describe that as unusual.

(Apologies on the table formatting - the column numbers just won't line up without the dots.)

And note that in Lotto Architect the user can create Draw Filter algorithms to allow study of numbers drawn for any number of requisite past draws - other than the draw filters pre-provided with the program.

regards

relowe

Hi Mr LA Man,

Can you confirm my thinking on the use of these Bernoulli tables please.

My usual use of these tables is to observe for balls to avoid ie what is the chance of appearance at the next draw after the ball has already appeared so many times in the last x draws. (And if it is below say 1% I avoid such a ball.) However if I observe that a ball has not been drawn for say 6 draws (in a 6 from 45 type lottery) the odds of it being drawn at the next draw ie the potential 7th draw of delay for the ball, are at some 39.55% (read from the zero times draw column at 6 draws and observe in the 1 times drawn column at draw 7 = 39.55%) Now given that a single ball in such a lottery has a chance of appearance of 13.3% does this mean that if it has not been drawn in 6 draws it has virtually three times the odds of appearing at the 7th draw?

Am I interpreting this correctly?

Many thanks

And incidentally thanks for the run down in another thread on your proposals for version 2.3 of Lotto Architect. It looks substantial, comprehensive and terrific. Can't wait.

relowe

This is the exact interpretation problem I mentioned in the previous post.

The Bernoulli tables show the following: "What is the chance of a number to appear x times in the last y draws".

Thus, this comment you mention

has an error in logic. What this 1% shows here (we are in decreasing ratio in that column) is a rather good indication that we should expect the number to appear actually. Just observe for example, the column 0. It starts with a high ratio and gradually, reduces to almost 0%, after e.g. 20 draws. This is totally normal, as we obviously expect the number to appear at least sometimes within the last 20 draws, and thus this is why we have such a low rate at 20 draws. Obviously, if we haven't a number in the last 20 draws, then on 21 draw (in column 0) we have an even smaller ratio, closer to 0%. That simply means, the chance for this number NOT to appear again (column 0) is almost 0%!!! In this situation, if you notice the column 1, it still has a low rate (but higher than column 0). This is still normal as statistically the number should have appeared not only once but more times. I would consider this number for inclusion for all the following draws actually. We do not remove this number, we include this number in our selection as it definitely owes to appear! These are extreme cases to observe but if we are in this position, very low %, then we definitely expect the number to appear (thus we move to the next higher columns to keep up with statistics results). The usual transition from one column to the next one is expected to occur at the highest ratios displayed on the next column. This is the comment I made to observe the column x+1, at position y+1 if the number has appeared x times in the last y draws.My usual use of these tables is to observe for balls to avoid ie what is the chance of appearance at the next draw after the ball has already appeared so many times in the last x draws. (And if it is below say 1% I avoid such a ball.)

On the other hand, if we face 1% or less, whilst in that column we are in increasing order, then we can easily ommit this number from our selection. Thus, take care of interpreting this 1% limit. Interpretation works differently, depending on if we are in increasing order or decreasing order. In decreasing order, we include the number, in increasing, we can ommit it. Probably this is my fault as I din't mention the difference of interpretaion of increasing/decreasing order in the initial posts.

Now, this comment

refers to percentage p, used by the Bernoulli tables. Thus, comparing these ratios is not valid, as they show different things. They do not conflict with each other of course! Conclusion is that we do not have three times the odds of appearing at the 7th draw in your example.Now given that a single ball in such a lottery has a chance of appearance of 13.3% ...

Mr LA,

Thanks for your detailed reply. Clearly I have not been using the tables as well as I could have, but I now expect to be able do so. Not too difficult to actually produce the table, but using it properly....

Thanks again - appreciated

relowe