Fun Facts with Quick Picks
Mike Sharkey
Fun Facts with Lottery Quick Picks
(The following was developed after a couple of email conversations
with a visitor to this group, who had some questions she felt I could answer.)
If you happened to have several tons of money lying about that you donıt
know WHAT to do with, you might try the following experiment.
Now, it is pretty well known that in a 6/49 lottery, there are 13,983,816
separate and distinct combinations. And it is also well known that, if
you were willing to spend the $$, time and effort, you could cover all
13,983,816 combinations using only and exactly 13,983,816 tickets.
But what would happen if you chose to purchase 13,983,816 Quick Picks??
How many duplicates would you get??
Or put another way, if that many QPs were purchased by people scattered
hither and yon, how likely would it be that the lottery would rollover?
Or, if the lottery was won, how likely would it be that the pot would have
to be split?
Well, not to go -too-far into the mathematics (Iım pretty sure its right,
Iım just not too sure that I could adequately explain it), but the
following numbers should be illustrative.
If 13,983,816 tickets were purchased using Quick Picks, you could
accurately expect that 63.21% of the possible combinations would be
drawn. Plus or minus about 0.04%
IOW, there should be 8,839,458 unique combinations ( plus or minus about
5,400 )
This means there will be 5,144,358 tickets that will duplicate at least
one other (existing) ticket.
But some of the combinations will be duplicated more than once! In fact,
you can reasonably expect that there will be 3,251,855 combinations that
will be duplicated once or more.
Letıs put together a little chart: For your 13,983,816 tickets what might
you expect?
Well, you could expect:
5,587,603 combinations covered by a single ticket each
5,587,683 tickets
2,055,564 combinations covered by two tickets 4,111,128 tickets
756,200 combinations covered by three tickets 2,268,600 tickets
278,190 combinations covered by four tickets 1,112,760 tickets
102,340 combinations covered by five tickets 511,700 tickets
37,649 combinations covered by six tickets 225,894 tickets
13,850 combinations covered by seven tickets 96,950 tickets
5,095 combinations covered by eight tickets 40,760 tickets
1,874 combinations covered by nine tickets 16,866 tickets
690 combinations covered by ten tickets 6,900 tickets
254 combinations covered by eleven tickets 2,794 tickets
93 combinations covered by twelve tickets 1,116 tickets
34 combinations covered by thirteen tickets 442 tickets
13 combinations covered by fourteen tickets 182 tickets
4 (or 5) combinations covered by fifteen tickets 60 tickets
1 (or 2) combinations covered by sixteen tickets ?
and (perhaps) one combination covered by more than sixteen
tickets ?
8,839,458 combinations covered by - - - - 13,983,835 tickets
(there is some error due to rounding , of course)
So what can we make of this? Well, not a whole lot, maybe - but it can be
sort of fun to juggle the numbers.
For example, it is 36.79 % likely that there will be a rollover in this
imaginary lottery.
On the other side of the coin, there is very close to 1 chance in 10,000
that the lottery prize will need to be split up among twelve or more
individual ticket holders.
Also, if the lottery IS won, the odds are almost exactly 3:2 that the
prize will have to be split up among more than one ticket holder.
Anyway, thrown out for general discussion, disparagement, discovery.
Any additions, alterations, comments, complaints or compliments cheerfully
received.
Mike Sharkey
With a little bit o' luck, wi' a lil' bit o' luck...
Ion Saliu
opposite angle as well. How many tickets of quick-picks are needed in
order to cover ALL 13,983,816 possible combinations? The answer can
only be obtained if we choose a degree of certainty (let's call it DC).
A relation that I call the "Funadamental Formula of Gambling" can give
us the answer:
N = log(1 - DC) / log(1 - p)
We can put it in words:
How many tickets (N) are needed to cover with 99% degree of certainty
(DC=.99) all combinations for an event of probability p = 1/13,983,816?
If we do the calculations, N needs to be 64,398,065. So, only if we buy
64,398,065 quick-picks there is a 99% chance that all 13,983,816
possible combinations are covered.
Ion Saliu
Sent via Deja.com http://www.deja.com/
Share what you know. Learn what you don't.
don mcdonald
> Date: Sun, 27 Jun 1999 14:06:07 GMT
> From: Mike Sharkey <msha...@localline.com>
> Newsgroups: rec.gambling.lottery
> Subject: Fun Facts with Quick Picks
>
>
> If 13,983,816 tickets were purchased using Quick Picks, you could
> accurately expect that 63.21% of the possible combinations would be
> drawn. Plus or minus about 0.04%
> IOW, there should be 8,839,458 unique combinations ( plus or minus about
> 5,400 )
>
> This means there will be 5,144,358 tickets that will duplicate at least
> one other (existing) ticket.
>
>
> Anyway, thrown out for general discussion, disparagement, discovery.
> Any additions, alterations, comments, complaints or compliments cheerfully
> received.
>
>
> Mike Sharkey
>
28.06.99 23:29 pm.
Let¹s put together a little chart: For your
13,983,816 tickets what might
you expect?
Well, you could expect:
5,587,603 combinations covered by a single ticket each
5,587,683 tickets
??###
typo.
I have done something similar. Using poisson distribution.
I am not sure, but our answers may be different.
And including Stephen Haslett?? Victoria Univ of Wellington n zeal.
2,055,564 combinations covered by two tickets 4,111,128 tickets
756,200 combinations covered by three tickets 2,268,600 tickets
278,190 combinations covered by four tickets 1,112,760 tickets
PROG LOTTO83shrk, POISSON DISTRIBUTION,
ASSUMING 13983816 49c6 POSSIBLE DIFFERENT
TICKETS., BY D. MCDONALD/ MATT GRAINGER
DOMINION, SUN 26.2.1989
NO. OF TICKETS, APPROX 14M , DAVID BALE / mike sharkey
EXPECTED NO. OF WINNERS = 1
X FACT P(X) % NO. OF SLOTS CUM. %
49 nCr 6 / (e * x!)
?? ?? ??
==========.
0 1 36.7879441 5144358.41 agree? 36.7879441
1 1 36.7879441 5144358.41 73.5758882
2 2 18.3939721 2572179.21 91.9698603
3 6 6.13132402 857393.069 98.1011843
4 24 1.532831 214348.267 99.6340153
5 120 0.306566201 42869.6535 99.9405815
6 720 0.0510943668 7144.94225 99.9916758
7 5040 7.29919526E-3 1020.70603 99.998975
8 40320 9.12399408E-4 127.588254 99.9998874
9 362880 1.01377712E-4 14.1764727 99.9999888
10 3628800 1.01377712E-5 1.41764727
ratio = 1/x = 1/10. 99.9999989
11 39916800 9.21615563E-7 0.128877025 99.9999999
12 479001600 7.68012969E-8 0.0107397521 99.9999999
13 6.2270208E9 5.90779207E-9 8.26134773E-4 99.9999999
14 8.71782912E10 4.21985148E-10 5.90096266E-5 99.9999999
15 1.30767437E12 2.81323432E-11 3.93397511E-6 99.9999999
16 2.09227899E13 1.75827145E-12 2.45873444E-7 99.9999999
17 3.55687428E14 1.03427732E-13 1.44631438E-8 99.9999999
18 6.40237371E15 5.74598513E-15 8.03507988E-10 99.9999999
19 1.216451E17 3.0242027E-16 4.22898941E-11 99.9999999
20 2.43290201E18 1.51210135E-17 2.11449471E-12 99.9999999
21 5.10909422E19 7.20048262E-19 1.00690224E-13 99.9999999
22 1.12400073E21 3.27294664E-20 4.57682836E-15 99.9999999
greetings mike, regards
/ don. (loto)
--
don.mcdonald | lot-to-it adviser + maths | rslts / 640 pers'lised # $5-
Pg. 329 W'ellington 2, new zealand | call mdr +64( 4) 389 6820.
formerly don.mc...@welcom.gen.nz, mcdon...@kosmos.wcc.govt.nz
R.eply-To: ds...@actrix.gen.nz
Michael Langran
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***************************************************
don mcdonald wrote:
> On Sun, 27 Jun 1999, Mike Sharkey wrote:
>
> > Date: Sun, 27 Jun 1999 14:06:07 GMT
> > From: Mike Sharkey <msha...@localline.com>
> > Newsgroups: rec.gambling.lottery
> > Subject: Fun Facts with Quick Picks
> >
> >
> > If 13,983,816 tickets were purchased using Quick Picks, you could
> > accurately expect that 63.21% of the possible combinations would be
> > drawn. Plus or minus about 0.04%
> > IOW, there should be 8,839,458 unique combinations ( plus or minus about
> > 5,400 )
> >
> > This means there will be 5,144,358 tickets that will duplicate at least
> > one other (existing) ticket.
> >
> >
> > Anyway, thrown out for general discussion, disparagement, discovery.
> > Any additions, alterations, comments, complaints or compliments cheerfully
> > received.
> >
> >
> > Mike Sharkey
> >
> 28.06.99 23:29 pm.
>
> Let零 put together a little chart: For your
--
Regards,
Michael Langran
***************************************************
GlobaLotto: The Internet's Online Lottery Resource
Michael Langran Phone: 972-490-7807
7050 Arapaho Rd. Suite 2039 Fax: 972-490-7734
Dallas, TX 75248 E-Mail: mlan...@airmail.net
http://www.globalottoplayers.com
ICQ#: 32349913 - mlangran
***************************************************