Even and Odd

[Stone]

No. od/ev perms expected% actual% expected actual
1=odd/0=even

1 000000 96909120 0.9625 0.8993 | 16.1 15
2 100000 127512000 1.2665 1.4988 | 21.1 25
3 010000 127512000 1.2665 1.3789 | 21.1 23
4 110000 153014400 1.5198 1.4388 | 25.3 24
5 001000 127512000 1.2665 1.0791 | 21.1 18
6 101000 153014400 1.5198 1.3789 | 25.3 23
7 011000 153014400 1.5198 1.9784 | 25.3 33
8 111000 167587200 1.6645 1.3789 | 27.8 23
9 000100 127512000 1.2665 1.0791 | 21.1 18
10 100100 153014400 1.5198 1.9185 | 25.3 32
11 010100 153014400 1.5198 1.8585 | 25.3 31
12 110100 167587200 1.6645 1.4988 | 27.8 25
13 001100 153014400 1.5198 1.9185 | 25.3 32
14 101100 167587200 1.6645 1.8585 | 27.8 31
15 011100 167587200 1.6645 2.0983 | 27.8 35
16 111100 167587200 1.6645 1.6787 | 27.8 28
17 000010 127512000 1.2665 0.8393 | 21.1 14
18 100010 153014400 1.5198 1.3189 | 25.3 22
19 010010 153014400 1.5198 1.4988 | 25.3 25
20 110010 167587200 1.6645 1.6787 | 27.8 28
21 001010 153014400 1.5198 1.2590 | 25.3 21
22 101010 167587200 1.6645 1.3789 | 27.8 23
23 011010 167587200 1.6645 1.3789 | 27.8 23
24 111010 167587200 1.6645 2.0983 | 27.8 35
25 000110 153014400 1.5198 1.8585 | 25.3 31
26 100110 167587200 1.6645 1.6787 | 27.8 28
27 010110 167587200 1.6645 1.5588 | 27.8 26
28 110110 167587200 1.6645 1.7986 | 27.8 30
29 001110 167587200 1.6645 1.8585 | 27.8 31
30 101110 167587200 1.6645 1.1990 | 27.8 20
31 011110 167587200 1.6645 1.6787 | 27.8 28
32 111110 153014400 1.5198 1.6787 | 25.3 28
33 000001 127512000 1.2665 1.6187 | 21.1 27
34 100001 153014400 1.5198 1.7386 | 25.3 29
35 010001 153014400 1.5198 1.6187 | 25.3 27
36 110001 167587200 1.6645 1.7986 | 27.8 30
37 001001 153014400 1.5198 1.6787 | 25.3 28
38 101001 167587200 1.6645 1.5588 | 27.8 26
39 011001 167587200 1.6645 1.3789 | 27.8 23
40 111001 167587200 1.6645 0.9592 | 27.8 16
41 000101 153014400 1.5198 1.9185 | 25.3 32
42 100101 167587200 1.6645 1.3189 | 27.8 22
43 010101 167587200 1.6645 1.0192 | 27.8 17
44 110101 167587200 1.6645 1.4388 | 27.8 24
45 001101 167587200 1.6645 1.0192 | 27.8 17
46 101101 167587200 1.6645 1.7986 | 27.8 30
47 011101 167587200 1.6645 1.4388 | 27.8 24
48 111101 153014400 1.5198 2.0983 | 25.3 35
49 000011 153014400 1.5198 1.2590 | 25.3 21
50 100011 167587200 1.6645 1.6787 | 27.8 28
51 010011 167587200 1.6645 1.5588 | 27.8 26
52 110011 167587200 1.6645 1.0192 | 27.8 17
53 001011 167587200 1.6645 1.7986 | 27.8 30
54 101011 167587200 1.6645 2.1583 | 27.8 36
55 011011 167587200 1.6645 2.0983 | 27.8 35
56 111011 153014400 1.5198 1.8585 | 25.3 31
57 000111 167587200 1.6645 1.4388 | 27.8 24
58 100111 167587200 1.6645 1.7986 | 27.8 30
59 010111 167587200 1.6645 1.4988 | 27.8 25
60 110111 153014400 1.5198 1.7386 | 25.3 29
61 001111 167587200 1.6645 1.4988 | 27.8 25
62 101111 153014400 1.5198 1.4388 | 25.3 24
63 011111 153014400 1.5198 1.9185 | 25.3 32
64 111111 127512000 1.2665 1.1391 | 21.1 19

 

[PAB]

Hi Stone,

Thanks for the information, I will have a look at it in more detail later.
If you used VBA code to calculate the information and table could you please post it for me.

Thanks in advance,
PAB

 

[Stone]

Hi Stone,

Thanks for the information, I will have a look at it in more detail later.
If you used VBA code to calculate the information and table could you please post it for me.

Thanks in advance,
PAB

Hi PAB, what I was looking for was whether or not the sequence was dependant on the fact that the pool varied as each number was taken out. Your suggestion of the coins being stacked up gave me an idea to test against (although using the built in rng), my conclusion is that it does affect what's drawn to some extent but only in the same way already seen from the distribution of possible sequences. So it's all just been another way of looking at the same thing.

It's not VB but here's the bit of code to work out the number of possible permutations that give each even odd sequence. It takes a little while as there's 10 billion of them and could be made faster by checking for duplicates outside the loops instead.

dim as integer odds(0 to 63)

for b1 as integer=1 to 49
for b2 as integer=1 to 49
for b3 as integer=1 to 49
for b4 as integer=1 to 49
for b5 as integer=1 to 49
for b6 as integer=1 to 49
dim as integer b(1 to 6)
b(1)=b1
b(2)=b2
b(3)=b3
b(4)=b4
b(5)=b5
b(6)=b6
dim as integer fail
for i as integer=1 to 5
for j as integer=i+1 to 6
if b(i)=b(j) then fail=1
next
next
if fail=0 then
dim as integer bits
for i as integer=1 to 6
bits or=(b(i) and 1)shl(i-1)
next
odds(bits)+=1
end if
next
next
next
next
next
next

 

[Stone]

This'll do it quite a bit quicker:

for b1 as integer=1 to 49
for b2 as integer=1 to 49
if (b1<>b2) then
for b3 as integer=1 to 49
if (b3<>b1)and(b3<>b2) then
for b4 as integer=1 to 49
if (b4<>b1)and(b4<>b2)and(b4<>b3) then
for b5 as integer=1 to 49
if (b5<>b1)and(b5<>b2)and(b5<>b3)and(b5<>b4) then
for b6 as integer=1 to 49
if (b6<>b1)and(b6<>b2)and(b6<>b3)and(b6<>b4)and(b6<>b5) then

odds((b1 and 1)or((b2 and 1)shl 1)or((b3 and 1)shl 2)or((b4 and 1)shl 3)or((b5 and 1)shl 4)or((b6 and 1)shl 5))+=1

end if
next
end if
next
end if
next
end if
next
end if
next
next

Stone

 

[Stone]

DOH!, just realised that can be worked out just be dividing the numbers into 2 groups (24 even and 25 odd) and just multiplying by the number of numbers left as 1 number from a group is used.

Stone

 

[PAB]

Hi Stone,

I hope you had a great Christmas.
I don't quite follow you, can you please post an example.

Regards,
PAB

 

[Stone]

Hi PAB, Christmas was ok thanks, hope you had a good one too.

for example

EOEEOE

E starts off as 24 and O as 25 then as each is used subtract 1 from it

(E=24)(O=25)(E=23)(E=22)(O=24)(E=21)

24*25*23*22*24*21=153014400

Stone

 

[PAB]

Hi Stone,

I will have a look at maybe building a formula or writing some code to produce the results like those in your table.

Regards,
PAB


----------------------------------------------------------------------------------------------------------------------------
Mathematics is the language of nature.
Everything around us can be represented and understood through numbers.
If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature.

 

[Stone]

Hi PAB, I was just kicking myself after realising there was a quicker way than looping through every permutation. Something like this would do it:

function even_odd_permutations(byval sequence as integer)as integer

'sequence=0-63
'bit0 of sequence=1st number even/odd (0=even 1=odd)
'bit1 of sequence=2nd number even/odd
'bit2 ............3rd

dim as integer permutations=1,eo(0 to 1)
eo(0)=24
eo(1)=25

for i as integer=0 to 5
permutations*=eo((sequence shr i)and 1)
eo((sequence shr i)and 1)-=1
next

function=permutations

end function

Stone

 

[PAB]

Hi Stone,

Happy New Year to you.
Have you had any luck with your quest?

Regards,
PAB


-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-
12:45, restate my assumptions.
Mathematics is the language of nature.
Everything around us can be represented and understood through numbers.
If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature.

 

[Stone]

Happy new year PAB and all,

LOL, no luck whatsoever

 

[Stone]

Hi there, been having a look at some things to do with even and odd. I dont know much about stats so don't really know what to make of this.

This is from the uk lottery

Typically the appearance of 3 even and 3 odd is something around 1:3.2

When looking at last 8 draws and the pattern is

not 3+3 (last draw)
not 3+3
not 3+3
3+3
not 3+3
not 3+3
not 3+3
3+3 (8 draws ago)

which has appeared 14 times, the draw after that has had 3+3 8 times and not 3+3 only 6 times.

I know the numbers are pretty low but is this just an anomoly that would be expected?

 

[Frank]

Hi there, been having a look at some things to do with even and odd. I dont know much about stats so don't really know what to make of this.

This is from the uk lottery

Typically the appearance of 3 even and 3 odd is something around 1:3.2

When looking at last 8 draws and the pattern is

not 3+3 (last draw)
not 3+3
not 3+3
3+3
not 3+3
not 3+3
not 3+3
3+3 (8 draws ago)

which has appeared 14 times, the draw after that has had 3+3 8 times and not 3+3 only 6 times.

I know the numbers are pretty low but is this just an anomoly that would be expected?

Hi Stone,

You are clearly ignoring the bonus ball to talk in terms of 3/3. In the UK lottery, the theoretical probability of a 3/3 even/odd split is 0.3329, which gives odds of (NOT 3/3) to (3/3) of (1-0.3329)/0.3329= 2.004 to 1 over the LIFETIME of draw results where LIFETIME is many thousands of results.

The ACTUAL number of 3/3 even/odds in the MAIN balls so far (1760 draws) is 543 draws. This is 30.85% of the whole lifetime so far draw history. This would calculate to actual odds (so far) of (1-0.3085)/0.3085 ]= 2.24 to 1.

I don't know where you got 1:3.2 from, as (3/3) to (NOT 3/3) ratio ?

I disagree with your assessment of the past 8 draws. There was a 3/3 Five draws ago and Nine draws ago, 1756 and 1752 ??

The whole point of statistics are that they apply on average to the whole of a population. You can't snip off a few draws and test them against the maths for the whole lifetime and look for a significant pattern. There is no pattern, it is random. So there is no anomaly and it may be another 10 years before the actual count of 3/3's over draw history approaches the theroetical figure.

 

[Stone]

Hi, I was meaning a (3/3) draw appears on average 1 in 3.2 draws, around 31% of the time. The pattern shown was compared throughout the entire draw history and it occurs 14 times with the next draw being (3/3) around 57% of the time (8 times out of the 14).

 

[Frank]

Hi, I was meaning a (3/3) draw appears on average 1 in 3.2 draws, around 31% of the time. The pattern shown was compared throughout the entire draw history and it occurs 14 times with the next draw being (3/3) around 57% of the time (8 times out of the 14).

Hi Stone,

Thank you for clarifying your figures, yes I understand what you are saying now. It is possible using both the theoretical odds and the actual odds to predict how likely what you describe can be.
However if we are not careful we can get deep into combinatorics, and I don't want to go down that road as I don't think I'm qualified to do a full sequence analysis. I like to take the simple approach which to my mind goes like this:-
First if we take the theoretical probabilty of a 3/3 :- 0.3329 and the theoretical probability of NOT 3/3 = 0.6671 and work out the combined probability of your sequence (note the order doesn't matter - just how many of each type in 8 draws ). Theres six NOTs and two 3/3's. so P= (0.6671)^6 x (0.3329)^2 =0.00967. There are 1753 opportunities of sequences of 8 in 1760 draws. So you would expect 1753 x 0.00967 sequences containing six NOTS and two 3/3s = 17.12. Thats the theory.

Using actual measured values of P=0.3085 from the results and making the same calculation we get a probability of six NOTS and two 3/3's in 8 draws of 0.0104. This works out at 18.24 such sequences.

You are getting 14 which seems fine to me looking at the above.

 

[PAB]

Hi Frank, I have tried sending you 3 emails and they keep bouncing back, I just thought I would let you know!

Regards,
PAB


-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-
12:45, restate my assumptions.
Mathematics is the language of nature.
Everything around us can be represented and understood through numbers.
If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature.

 

[Frank]

Hi PAB,

I received the one you sent on 8 Nov which contained the run times for the macros. None since. I've tested the address by sending myself mail from another domain and it recieved OK. My Spam filter shows I received mail from various spammers every day since then, so I don't know what the problem might be.

 

[PAB]

Hi Frank,

I received the one you sent on 8 Nov which contained the run times for the macros. None since. I've tested the address by sending myself mail from another domain and it recieved OK. My Spam filter shows I received mail from various spammers every day since then, so I don't know what the problem might be.

VERY strange!
FOUR emails were bounced back yesterday by Postmaster.
Anyway, you should have received one from me now replying to yours which seems to have worked.

Regards,
PAB


-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-
12:45, restate my assumptions.
Mathematics is the language of nature.
Everything around us can be represented and understood through numbers.
If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature.

 

[Stone]

Hi Stone,

Thank you for clarifying your figures, yes I understand what you are saying now. It is possible using both the theoretical odds and the actual odds to predict how likely what you describe can be.
However if we are not careful we can get deep into combinatorics, and I don't want to go down that road as I don't think I'm qualified to do a full sequence analysis. I like to take the simple approach which to my mind goes like this:-
First if we take the theoretical probabilty of a 3/3 :- 0.3329 and the theoretical probability of NOT 3/3 = 0.6671 and work out the combined probability of your sequence (note the order doesn't matter - just how many of each type in 8 draws ). Theres six NOTs and two 3/3's. so P= (0.6671)^6 x (0.3329)^2 =0.00967. There are 1753 opportunities of sequences of 8 in 1760 draws. So you would expect 1753 x 0.00967 sequences containing six NOTS and two 3/3s = 17.12. Thats the theory.

Using actual measured values of P=0.3085 from the results and making the same calculation we get a probability of six NOTS and two 3/3's in 8 draws of 0.0104. This works out at 18.24 such sequences.

You are getting 14 which seems fine to me looking at the above.

Hi Frank, what I was trying to get at is that on the 14 occasions that pattern appears, the draw after results in (3/3) 8 out of the 14 which is more than I would expect but is something like that expected due to the low number of occurences?

 

[PAB]

Hi Stone,

...what I was trying to get at is that on the 14 occasions that pattern appears, the draw after results in (3/3) 8 out of the 14 which is more than I would expect...

Sorry to jump in here.

Currently, as at Draw 1,763 for Wednesday 14 Nov 2012 for the UK Lotto the Odd & Even drawn numbers stand at:-

Average Odd & Even Numbers are respectively 3.07 & 2.93.

Median Odd & Even Numbers are respectively 3 & 3.

This is really what is expected with regard to the number of draws concerned.
I do think you're right though that if we monitor these on a draw to draw basis, that if a particular grouping is well overdue that it might be worth compiling combinations that account for this fact.

As you probably know, the total Expected combinations for ALL C(49,6) Odd & Even numbers are as follows:-

Distribution Combinations Percent 1 in Every
0 Odd + 6 Even 134,596 0.96% 103.89
1 Odd + 5 Even 1,062,600 7.60% 13.16
2 Odd + 4 Even 3,187,800 22.80% 4.39
3 Odd + 3 Even 4,655,200 33.29% 3.00
4 Odd + 2 Even 3,491,400 24.97% 4.01
5 Odd + 1 Even 1,275,120 9.12% 10.97
6 Odd + 0 Even 177,100 1.27% 78.96
Totals > 13,983,816 100.00%

The total combinations for the Expected Odd & Even numbers to date are as follows:-

Distribution Expected Percent 1 in Every
0 Odd + 6 Even 16.97 0.96% 103.89
1 Odd + 5 Even 133.97 7.60% 13.16
2 Odd + 4 Even 401.90 22.80% 4.39
3 Odd + 3 Even 586.90 33.29% 3.00
4 Odd + 2 Even 440.18 24.97% 4.01
5 Odd + 1 Even 160.76 9.12% 10.97
6 Odd + 0 Even 22.33 1.27% 78.96
Totals > 1,763.00 100.00%

The total combinations for the Actual Odd & Even numbers to date are as follows:-

Distribution Actual Percent 1 in Every
0 Odd + 6 Even 16 0.91% 110.19
1 Odd + 5 Even 129 7.32% 13.67
2 Odd + 4 Even 436 24.73% 4.04
3 Odd + 3 Even 545 30.91% 3.23
4 Odd + 2 Even 429 24.33% 4.11
5 Odd + 1 Even 189 10.72% 9.33
6 Odd + 0 Even 19 1.08% 92.79
Totals > 1,763 100.00%

This gives as a difference of the Expected and Actual of:-

Distribution Difference Percent 1 in Every
0 Odd + 6 Even -0.97 -0.05% 6.29
1 Odd + 5 Even -4.97 -0.28% 0.51
2 Odd + 4 Even 34.10 1.93% -0.34
3 Odd + 3 Even -41.90 -2.38% 0.23
4 Odd + 2 Even -11.18 -0.63% 0.10
5 Odd + 1 Even 28.24 1.60% -1.64
6 Odd + 0 Even -3.33 -0.19% 13.83
Totals > 0.00 0.00%

Frank actually gave a very good analysis and interpretation in his previous post in this thread.

Regards,
PAB


-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-∏-
12:45, restate my assumptions.
Mathematics is the language of nature.
Everything around us can be represented and understood through numbers.
If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature.