A Study Of Bankers

By Paul McCoy / The Lottery Mine

```Bankers or "Key" numbers are a popular way to play the lottery. It is an
attractive strategy for a variety of reasons, but chief among them is
probably the fact that it allows the player to both play a larger field
of numbers and reduce the cost of play at the same time. If the
designated bankers prove to be correct, the payoff can be huge...usually
with multiple matches of the lower tier prizes. If, on the other hand,
the bankers don't show, the results can be disastrous. That little word
"if" carries huge implications; and because of that it is important to
fully understand both the risks and the rewards when employing this
strategy. As is my usual custom, I am using a 649 lottery as the model
here. Bonus numbers are  ignored in order to keep a level playing field
since not all 649 lotteries draw a 7th bonus number.

The first scenario below is the most basic and straightforward example
of them all. It illustrates what happens when you choose any 1 number as
a banker and you are playing the full field of 49 numbers.

First of all, you end up with 1,712,304 lines broken down thusly:

Matches Lines   Relative Odds   Conditional Odds
1       962598  14.53           1.78
2       617050  22.66           2.77
3       123410  113.31          13.87
4       9030    1548.60         189.62
5       215     65041.00        7964.20
6       1       13983816        1712304

This table shows both the "relative" odds and the "conditional" odds of
matching the various prize levels. Both are dependent on *IF* the banker
number is correct. It is important to bear in mind that the *overall*
odds for any single line played remain constant. What we are dealing
with here is "relative odds" in addition to conditional odds. The odds
become relative whenever you define any subset of lines. In this case,
the subset consists of 1,712,304 lines that are defined by the fact that
one particular number appears in every one of them.

The constant overall odds of any random line matching 3 of the drawn
numbers is about 1 in 56.66. Notice the effect of committing yourself to
a banker number. The relative odds of any one of your lines from this
subset matching 3 of the drawn numbers is shown as 1 in 113.31. The odds
have doubled because exactly 50% of all the possible match3's have been
excluded from the subset. So, instead of working in your favour, the
playing of a banker is (in a sense) working *against* you. The same
holds true for the other lower tier matches. Only the odds for a jackpot
match remain the same. Now look at the conditional odds for a match3. If
the banker number is indeed correct, the odds for any of the lines being
a match 3 have dropped to 1 in 13.87. Again, the same holds true for the
other prize levels *including* the jackpot. If the condition is true,
the odds for a jackpot match have dropped from 1 in 13,983,816 to a
"mere" 1 in 1,712,304. That's the good news. The bad news is that if
your banker number is *not* drawn, you are immediately disqualified from
having a jackpot match...although you still have a chance at a maximum
of 5 matches and multiple lower prizes.

Of course this example is academic at best. No one can afford to play
1.7 million lines on a single draw. This would necessitate the use of an
abbreviated wheel design and probably some heavy filtering in order to
make this strategy reasonably affordable. The other alternative is that
the field of numbers would have to be drastically reduced. In either
case, it is my opinion that you may as well just play random lines to
begin with. Of course you could also play a normal wheel with a covering
design that guarantees a certain prize level within an affordable number
of lines.

What happens if you play more than 1 banker number? The following shows
the figures for 2,3 and 4 bankers. Again, this is using all 49 numbers.

2 Bankers
==========

Matches Lines   Relative Odds   Conditional Odds
2       123410  113.31          1.45
3       49364   283.28          3.61
4       5418    2580.99         32.92
5       172     81301.26        1037.01
6       1       13983816        178365

3 Bankers
==========

Matches Lines   Relative Odds   Conditional Odds
3       12341   1133.12         1.23
4       2709    5161.98         5.60
5       129     108401.67       117.67
6       1       13983816        15180

4 Bankers
==========

Matches Lines   Relative Odds   Conditional Odds
4       903     15485.95        1.10
5       86      162602.51       11.51
6       1       13983816        990

Notice how the relative odds get ever higher and the conditional odds
get ever lower, while the total number of lines gets increasingly
smaller. By the time we reach 4 bankers, we're down to only 990 lines.
If we took this one step further to 5 bankers, we'd be down to only 44

Now consider the following:

There are 49 ways to designate 1 banker...any 1 of which has a 1 in 8.17
chance of containing the jackpot numbers within it's subset of 1,712,304
lines. The banker must be correct or the possibility of a jackpot win is
lost, although the possibility of a match5 remains. A reduced field of
numbers played means a rapid reduction of favourable odds.

There are 1176 ways to choose 2 bankers...any 1 of which has a 1 in 78.4
chance of containing the jackpot numbers within it's subset of 178,365
lines. Both bankers must be correct or the best that can be hoped for is
a match4, although 1 correct still leaves a slim chance at a match5. A
reduced field of numbers played means a rapid reduction of favourable
odds.

There are 18,424 ways to choose 3 bankers...any 1 of which has a 1 in
921.2 chance of containing the jackpot numbers within it's subset of
15,180 lines. All 3 bankers must be correct or the best that can be
hoped for is a match3, although 2 or 1 correct still leaves slim chances
at matching 5 or 4 respectively. A reduced field of numbers played means
a rapid reduction of favourable odds.

There are 211,876 ways to choose 4 bankers...any 1 of which has a 1 in
14,125.07 chance of containing the jackpot numbers within it's subset of
990 lines. Unless at least 3 bankers are correct, you're definitely
going to lose money on this one. Your name had better be Kreskin or you
had better have horseshoes up the yin-yang before flirting with this. A
reduced field of numbers played means a rapid reduction of favourable
odds.

Given the above facts, do any of these appear to be particularly
attractive? Each becomes increasingly affordable while also becoming
increasingly risky. At what point do we see any kind of acceptable
balance between cost and risk? If there is such a point, I for one fail
to see it. Feel free to illuminate both myself and the rest of the
readers here if you see things differently. That's why we're here isn't
it? I hope everyone realizes that my motive here is to promote serious
thought and discussion rather than simply trying to impose my views on
the rest of the world.

AFTERTHOUGHTS
================

Players around the world win huge amounts of money almost every day
without so much as a second thought in regard to strategy of any
description. I sometimes feel that there is a penalty to pay when you
think about something too much. The harder you try...the more elusive
does the goal become. This is especially true when you begin to believe
that there is some "secret" method that will grant you the keys to the
kingdom. Those "secrets" don't exist. As a wise rabbit once said, "Trix
are for kids".

Anyone else around here interested in dealing with reality? Or has this
become nothing more than a haven for flights of pure fantasy...where
desperate people are clamouring and clawing for any *shred* of
information (right or wrong) that will miraculously *give* them the key
to untold riches? Don't bet on it! Do your own work with the knowledge
and tools that have been provided by the hard work of others. *Read* the
rgl FAQ which is frequently posted here. Visit the websites and check
out the software that it will point you to. They are there for a good
reason. Particular merit should be awarded to:

Mr. John Rawson for Covermaster

There are others that could be mentioned and I don't mean to offend
anyone by an omission from the *very* short list that appears above.

Paul McCoy
IXL Software

Visit Paul at . . .
The Lottery Mine archived copy
Home Of LDM - Sensible Shareware At A Sensible Price
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