Bernoulli Table for Canadian649

[Grandmaster]

Inspired by LottoArchitect work with Bernoulli tables

PROBABILITY OF A NUMBER TO APPEAR 01234567 TIMES IN THE LAST N DRAWS

1 85.71% 14.29% #NUM! #NUM! #NUM! #NUM! #NUM! #NUM!
2 73.47% 24.49% 2.04% #NUM! #NUM! #NUM! #NUM! #NUM!
3 62.97% 31.49% 5.25% 0.29% #NUM! #NUM! #NUM! #NUM!
4 53.98% 35.99% 9.00% 1.00% 0.04% #NUM! #NUM! #NUM!
5 46.27% 38.56% 12.85% 2.14% 0.18% 0.01% #NUM! #NUM!
6 39.66% 39.66% 16.52% 3.67% 0.46% 0.03% 0.00% #NUM!
7 33.99% 39.66% 19.83% 5.51% 0.92% 0.09% 0.01% 0.00%
8 29.14% 38.85% 22.66% 7.55% 1.57% 0.21% 0.02% 0.00%
9 24.97% 37.46% 24.97% 9.71% 2.43% 0.40% 0.04% 0.00%
10 21.41% 35.68% 26.76% 11.89% 3.47% 0.69% 0.10% 0.01%
11 18.35% 33.64% 28.03% 14.02% 4.67% 1.09% 0.18% 0.02%
12 15.73% 31.45% 28.83% 16.02% 6.01% 1.60% 0.31% 0.04%
13 13.48% 29.21% 29.21% 17.85% 7.44% 2.23% 0.50% 0.08%
14 11.55% 26.96% 29.21% 19.47% 8.92% 2.97% 0.74% 0.14%
15 9.90% 24.76% 28.89% 20.86% 10.43% 3.82% 1.06% 0.23%
16 8.49% 22.64% 28.30% 22.01% 11.92% 4.77% 1.46% 0.35%
17 7.28% 20.62% 27.49% 22.91% 13.36% 5.79% 1.93% 0.51%
18 6.24% 18.71% 26.51% 23.56% 14.73% 6.87% 2.48% 0.71%
19 5.35% 16.93% 25.39% 23.98% 15.99% 7.99% 3.11% 0.96%
20 4.58% 15.27% 24.18% 24.18% 17.13% 9.14% 3.81% 1.27%
21 3.93% 13.75% 22.91% 24.18% 18.14% 10.28% 4.57% 1.63%
22 3.37% 12.34% 21.60% 24.00% 19.00% 11.40% 5.38% 2.05%
23 2.89% 11.06% 20.28% 23.66% 19.72% 12.49% 6.24% 2.53%
24 2.47% 9.89% 18.96% 23.18% 20.28% 13.52% 7.14% 3.06%
25 2.12% 8.83% 17.67% 22.57% 20.69% 14.48% 8.05% 3.64%
26 1.82% 7.87% 16.40% 21.87% 20.96% 15.37% 8.97% 4.27%
27 1.56% 7.01% 15.19% 21.09% 21.09% 16.17% 9.88% 4.94%
28 1.34% 6.23% 14.02% 20.25% 21.09% 16.87% 10.78% 5.65%
29 1.14% 5.53% 12.91% 19.36% 20.97% 17.48% 11.65% 6.38%
30 0.98% 4.90% 11.85% 18.44% 20.74% 17.98% 12.48% 7.13%
31 0.84% 4.34% 10.86% 17.50% 20.41% 18.37% 13.27% 7.90%
32 0.72% 3.84% 9.93% 16.55% 19.99% 18.66% 14.00% 8.66%
33 0.62% 3.40% 9.06% 15.60% 19.50% 18.85% 14.66% 9.43%
34 0.53% 3.00% 8.25% 14.67% 18.95% 18.95% 15.26% 10.17%
35 0.45% 2.65% 7.50% 13.75% 18.33% 18.95% 15.79% 10.90%
36 0.39% 2.33% 6.81% 12.86% 17.68% 18.86% 16.24% 11.60%
37 0.33% 2.06% 6.17% 11.99% 16.99% 18.69% 16.61% 12.26%
38 0.29% 1.81% 5.58% 11.16% 16.28% 18.45% 16.91% 12.88%
39 0.24% 1.59% 5.04% 10.36% 15.55% 18.14% 17.13% 13.46%
40 0.21% 1.40% 4.55% 9.60% 14.81% 17.77% 17.27% 13.98%
41 0.18% 1.23% 4.10% 8.88% 14.06% 17.34% 17.34% 14.45%
42 0.15% 1.08% 3.69% 8.20% 13.32% 16.87% 17.34% 14.87%
43 0.13% 0.95% 3.32% 7.55% 12.59% 16.37% 17.28% 15.22%
44 0.11% 0.83% 2.98% 6.95% 11.87% 15.83% 17.15% 15.51%
45 0.10% 0.73% 2.67% 6.38% 11.17% 15.26% 16.96% 15.75%
46 0.08% 0.64% 2.39% 5.85% 10.48% 14.68% 16.72% 15.92%
47 0.07% 0.56% 2.14% 5.36% 9.82% 14.08% 16.42% 16.03%
48 0.06% 0.49% 1.92% 4.90% 9.18% 13.47% 16.09% 16.09%
49 0.05% 0.43% 1.71% 4.47% 8.57% 12.86% 15.72% 16.09%
50 0.04% 0.37% 1.53% 4.08% 7.99% 12.25% 15.31% 16.04%
51 0.04% 0.33% 1.36% 3.71% 7.43% 11.64% 14.87% 15.93%
52 0.03% 0.29% 1.22% 3.38% 6.90% 11.04% 14.41% 15.78%
53 0.03% 0.25% 1.08% 3.07% 6.39% 10.44% 13.93% 15.58%
54 0.02% 0.22% 0.96% 2.79% 5.92% 9.87% 13.43% 15.35%
55 0.02% 0.19% 0.86% 2.53% 5.47% 9.30% 12.92% 15.07%
56 0.02% 0.17% 0.76% 2.29% 5.05% 8.76% 12.40% 14.77%
57 0.02% 0.15% 0.68% 2.07% 4.66% 8.23% 11.88% 14.43%
58 0.01% 0.13% 0.60% 1.87% 4.29% 7.72% 11.36% 14.06%
59 0.01% 0.11% 0.53% 1.69% 3.94% 7.23% 10.84% 13.68%
60 0.01% 0.10% 0.47% 1.52% 3.62% 6.76% 10.32% 13.27%
61 0.01% 0.08% 0.42% 1.37% 3.32% 6.31% 9.81% 12.85%
62 0.01% 0.07% 0.37% 1.24% 3.04% 5.88% 9.31% 12.42%
63 0.01% 0.06% 0.33% 1.11% 2.78% 5.48% 8.82% 11.97%
64 0.01% 0.06% 0.29% 1.00% 2.55% 5.09% 8.34% 11.52%
65 0.00% 0.05% 0.26% 0.90% 2.33% 4.73% 7.88% 11.07%

EXAMPLE:

Which is the probability of a number to appear 5 times in the last 10 draws for the Canadian649?

R=0.69%

Then as a tactic we could safely remove numbers that have appeared 5 or more times in the last 10 draws because it has a very low possibility to appear in next draw (6 in 11=0.18%).


Which is the probability of a number to appear 0 times in the last 65 draws for the Canadian649?

R=0.00%

Then as a tactic if a number has a skip of 65 or more we could choose these number, because it has a chance to appear soon.

 

[PAB]

Hi Grandmaster,

Very Interesting Table.
Are there Excel Formulas that are Able to Calculate the Table Above Please.

All the Best.
PAB

 

[GillesD]

Use of Excel for Bernouilli or Binomial distribution

PAB

You can get all the values given by Grandmaster in his table using the binomial distribution functions in Excel. Although I question the value he selected as the probability of success.

The function to use is BINOMDIST(nSucc,nTrials,pSucc,Cumul) where the various parameters required by this function are:
- nSucc is the number of successes to be obtained in the trials to be made;
- nTrials is the number of independent trials to be made;
- pSucc is the probability of success in each trial;
- Cumul is a TRUE/FALSE value depending whether you want a cumulative probability (True) or simply the probability of nSucc in nTrials (False).

Using the example given by Grandmaster, the parameters will be:
- nSucc = 5 since we are looking for 5 successes (or a number coming out in a draw);
- nTrials = 10 since we want those 5 successes to happen in the next 10 trials (or draws);
- pSucc = 14.29% (or 7/49); here I disagree with Grandmaster as I would rather use 12.24% (or 6/49). Both values can be good depending on your need; with 14,29% a success is obtained if the number comes out in the 7 numbers (including the Bonus number) but the 12.24% value applies only if the number comes out in the 6 regular numbers.
- Cumul = FALSE as we want the the probability of nSucc in nTrials

The function BINOMDIST(5,10,0,1429,FALSE) gives 0.69% while the function BINOMDIST(5,10,0,1224,FALSE) gives 0.36%.

Also the function BINOMDIST(5,10,0,1429,TRUE) gives 99.90% or the sum of 21.41, 35.68, 26.76, 11.89, 3.47 and 0.69 (the individual values for 0, 1, 2, 3, 4 and 5 successes).

 

[Grandmaster]

Hi,

Thank you GillesD for the explanation of the formula in excel, it is more easy than the formula I used


PAB,

I used a variation of the same formula as GillesD, C(y,x) * p^x * (1-p)^(y-x) =


Excel type:

7/49

p=COMBIN(48,6)/COMBIN(49,7)=0.142857143

=((COMBIN(10,5)*(0.142857143^5)*(1-0.142857143)^(10-5))*(100))

=0.69%


6/49

p=COMBIN(48,5)/COMBIN(49,6)=0.122448980

=((COMBIN(10,5)*(0.12244898^5)*(1-0.12244898)^(10-5))*(100))

=0.36%


NOTE: The formula I used was obtained by an explanation of Lottoarchitect regarding Bernoulli table in LA forum.

 

[PAB]

Hi GillesD,

I Posted the Following the Other Day, But for some Reason it did Not Appear.

Thank you for the Detailed Explanation, Excellent as Normal.
I have NOT Used the Function BINOMDIST(nSucc,nTrials,pSucc,Cumul) Before, it Seems Very Interesting. There are Probably Other Lotto Orientated Tasks that it could be Used for.
I Setup a Spreadsheet with the Total Numbers Drawn in Cell A1, the Total Numbers Drawn from in Cell A2, the "cumulative" in Cell A3, & the "probability_s" in Cell A4, it Works Great. I Changed the Total Drawn from 6 ( in Cell A1 ) to 7 and it Automatically Updated the Table.

Grandmaster,

Thanks for the Formulas and Explanation.

All the Best.
PAB

 

[PAB]

Grandmaster & GillesD,

What would the Table be if we were to Set the Parameter Cumul to TRUE.
Also, what would the Formulas be Please.
Using the Example Previously ( with the Cumul Set to FALSE ) of the Probability of a Number Appearing 5 Times in the Last 10 Draws in a 649 Lotto :-

p=COMBIN(48,5)/COMBIN(49,6)=0.122448980

=((COMBIN(10,5)*(0.12244898^5)*(1-0.12244898)^(10-5))*(100))

=0.36%

Basically, what would these Formulas be if the Cumul was TRUE Please.

Thanks in Advance.
All the Best.
PAB

 

[Grandmaster]

Hi PAB,

I will use the formulla of GillesD in excel for the Cumul set to TRUE:

=BINOMDIST(5,10,0.122448980,TRUE)
=1
=1*100=100%

(the individual values for 0, 1, 2, 3, 4 and 5 successes).

Regards

 

[Grandmaster]

This was an edit to previous post:

Hi PAB,

I will use the formulla of GillesD in excel for the Cumul set to TRUE:

=BINOMDIST(5,10,0.1224,TRUE)
=.999545
=1*0.999545=99.95%

(the individual values for 0, 1, 2, 3, 4 and 5 successes).

5 0.003610258
4 0.021561265
3 0.088298513
2 0.237302252
1 0.377925808
0 0.270846828
TOTAL=0.999544924=99.95%


Regards

 

[PAB]

Thanks for the Reply Grandmaster,

This is the Excel Formula that I am Using at Present for the Probability of a Number Appearing 5 Times in the Last 10 Draws in a 649 Lotto with the "cumulative" Set to "FALSE" :-

=(1-B47/B48)^(B60-H50)*(B47/B48)^H50*COMBIN(B60,H50)

This gives the Answer of 0.36%.

Cell "H50" is the Number of Times Appearing ( 5 Times ).
Cell "B60" is the Number of Draws ( 10 Draws ).
Cell "B47" is the Numbers Drawn ( 6 Numbers ).
Cell "B48" is the Numbers Drawn from ( 49 Numbers ).

What would the Above Formula look like with the Parameter Set to "TRUE".

I know it would be a LOT Easier if I was to Use the Formula …
=BINOMDIST(H50,B60,COMBIN(B48-1,B47-1)/COMBIN(B48,B47),TRUE)
… But I would Really like to know the Answer Using the NON BINOMDIST Function Way.

Thanks in Advance.
All the Best.
PAB

 

[Grandmaster]

… But I would Really like to know the Answer Using the NON BINOMDIST Function Way.

Hi PAB,

Maybe GillesD or Lottoarchitect could answer the question..

If I found the answer I'll tell you..

Regards

 

[PAB]

Thanks for the Reply Grandmaster,

Maybe GillesD or Lottoarchitect could answer the question.

I will have Another look at it Tomorrow.

All the Best.
PAB