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Lotto Analysis by Brandon MackayTRANSCRIPT
“Winning” Lotto Strategies 1
Analysis of “Winning” Lotto Strategies
Analysis of “Winning” Lotto Strategies
Brandon K. Mackay
Brigham Young University
“Winning” Lotto Strategies 2
Abstract
“Winning lotto strategies” are published abundantly in books and on the internet.
Ranging in approaches from calculating delta numbers to finding numbers that show up
in pairs, one can discover just about any method. This article analyzes a few such
strategies and identifies their fallacies.
Analysis of Winning Lotto Strategies
Everyone that plays a lottery hopes to win the big jackpot by finding a key that
will help them pick the winning numbers. Many people analyze statistics and pour over
tables of data in hopes of finding a winning strategy. Many of these people
unintentionally make mistakes in their assumptions and calculations, while others do so
on purpose to make their strategy appear superior and thereby sell their product. These
products can be anything from books to wheels to computer programs.
We will look at a few of the many different “winning strategy” claims and lotto
systems to identify their logical and mathematical fallacies. Then we will use the Oregon
State Lotto game, “Megabucks,” to show how lotteries are administered in an unbiased
way.
Strategies
Overdue Numbers
One common, subconscious strategy that is used looks for numbers that are
“overdue.” Suppose that while betting on heads/tails of coin flips, the most recent 15
flips have come up heads. This is a sign to many people that the probability of heads
appearing again is extremely small, so they place their bet on tails.
0
5
1015
20
25
30
3540
45
50
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46
Lotto Numbers
Tim
es S
elect
ed
“Winning” Lotto Strategies 3
On the previous page is a graph of winning numbers for the past 273 drawings.
Following this mindset, one would choose 11, 36, 46, or any combination of these,
claiming these numbers were overdue to be picked. In addition, hit/skip charts are
created by hand or on computers, to help keep track of how long it has been since certain
numbers have appeared. Usually, serious hit/skip charts span the last 30-50 drawings.
An abbreviated version with the numbers 1-10 for the last 15 drawings follows.
Number Last 15 Drawings 1 X X 2 X X X X 3 X X X4 X 5 X X X X 6 X X X 7 X X X8 X X X X X 9 X X X X
10
From the chart, a player might notice that the number 10 has not been drawn in
the last 15 drawings and is therefore, overdue to be drawn. A similar case could be made
for the number 4, which last came up 13 drawings ago.
The player might do this because he erroneously thinks that if the lotto is fair then
eventually each number will be chosen the same number of times. This common
misunderstanding is sometimes called the Law of Averages. In reality, if the lotto is fair,
each number will have the same probability of being selected on any given drawing. It
does not matter if the number 13 has been drawn ten times in a row. It will have the
same probability of being drawn on the next draw as any other number. In a game of 48
possible numbers, this probability is 1/48 (~ 2%). This is because each drawing is
independent of all past drawing. Bear in mind that the drawing device does not care
about past winning numbers; what happened in the past has no relevance on future
drawings. Thinking of events as dependant, when they are independent, is a common
error, and is officially called the “Gambler’s Fallacy.”
Sums of Winning Numbers
Another proposed “winning” strategy creates a distribution of the sums of the
winning lotto numbers. This system plots the sum of the winning numbers of each
“Winning” Lotto Strategies 4
drawing and then finds the mean of this distribution. Then, by choosing your lotto
numbers so that their sum is equal to the mean of the previous drawings, you supposedly
improve your chances. To illustrate this, let us look at the data from the last 273
drawings of the Oregon State Lotto game “Megabucks.”
Winning Numbers ∑7 14 18 29 30 32 130
3 6 14 22 26 28 99
3 6 14 24 26 28 101
12 16 24 29 34 46 161
2 18 23 30 31 46 150
8 15 22 28 32 46 151
8 24 26 28 37 45 168
1 14 17 18 26 43 119
1 14 25 30 38 44 152
2 5 27 34 40 45 153
11 14 18 36 41 42 162
2 5 9 14 34 47 111
3 15 31 44 45 48 186
9 15 17 18 22 23 104
5 20 31 32 36 47 171
5 9 18 24 38 39 133
1 7 26 29 37 45 145
18 19 34 37 38 42 188
3 5 23 29 36 38 134
2 6 23 38 43 46 158
4 16 18 21 40 45 144
20 22 28 37 42 46 195
1 2 16 27 33 34 113
14 15 25 38 39 42 173
3 4 23 31 42 44 147
6 10 15 26 42 47 146
6 9 12 18 24 40 109
8 12 13 15 30 43 121
12 18 24 27 32 47 160
6 13 17 20 25 27 108
13 14 17 31 35 43 153
9 15 19 21 29 44 137
2 7 15 29 37 42 132
11 21 27 28 36 38 161
20 23 24 27 32 35 161
1 4 5 14 27 31 82
4 18 24 34 37 47 164
15 25 29 35 38 47 189
12 14 18 20 36 37 137
15 20 25 38 42 43 183
8 14 22 26 27 29 126
4 11 14 32 38 45 144
11 21 43 44 46 47 212
7 12 21 33 38 39 150
5 15 17 38 40 46 161
5 11 17 20 26 46 125
10 14 21 29 33 35 142
4 8 9 13 23 29 86
1 27 30 34 36 40 168
7 13 14 17 25 42 118
4 7 14 15 40 45 125
11 19 24 29 37 39 159
5 12 16 23 34 38 128
4 10 17 18 24 26 99
1 2 5 24 27 47 106
8 12 17 24 27 48 136
8 13 26 34 44 48 173
31 35 36 37 40 42 221
9 12 24 30 31 47 153
10 20 38 43 44 47 202
3 11 18 26 39 48 145
8 10 12 27 39 42 138
6 20 25 38 39 41 169
1 10 11 15 42 45 124
11 13 23 35 43 46 171
13 14 18 24 41 48 158
2 17 32 33 43 44 171
1 12 26 34 38 44 155
3 15 16 28 33 37 132
1 5 13 24 40 46 129
5 7 9 21 34 47 123
1 13 28 33 46 48 169
10 15 24 35 39 43 166
4 15 28 37 44 45 173
10 31 37 43 44 47 212
1 14 16 24 25 33 113
16 22 23 24 27 33 145
10 22 25 32 44 46 179
1 4 8 15 20 48 96
12 14 17 18 29 37 127
1 3 4 16 22 37 83
8 19 29 40 41 44 181
8 16 23 29 38 42 156
4 11 15 18 31 45 124
2 6 11 21 23 31 94
2 12 13 14 26 48 115
5 34 35 37 43 44 198
25 26 28 31 41 47 198
7 9 16 19 42 46 139
11 14 15 24 31 48 143
3 6 10 12 34 45 110
12 26 27 32 35 45 177
16 18 21 44 45 46 190
9 13 21 32 41 47 163
12 26 34 37 45 48 202
4 7 9 10 23 26 79
2 3 13 14 21 25 78
3 9 11 15 41 46 125
15 21 25 42 44 47 194
15 17 29 31 35 38 165
3 4 6 27 30 41 111
2 6 28 31 37 41 145
7 9 13 26 31 47 133
13 27 33 34 40 44 191
1 6 11 12 19 40 89
3 14 18 40 43 46 164
16 22 38 40 42 48 206
8 22 23 27 40 45 165
2 6 12 13 35 39 107
10 13 14 26 40 43 146
5 12 20 29 38 45 149
4 21 31 32 34 48 170
3 6 13 37 38 45 142
8 25 32 33 35 41 174
2 27 31 33 43 44 180
6 9 13 25 40 43 136
6 7 11 12 32 47 115
6 14 16 27 32 40 135
6 15 24 30 39 44 158
14 28 38 39 42 44 205
1 2 6 23 26 44 102
10 12 23 35 39 41 160
18 21 22 27 30 46 164
1 8 21 24 27 38 119
5 6 14 19 21 30 95
3 18 26 29 33 48 157
1 2 10 14 15 17 59
5 13 20 35 38 39 150
2 13 15 16 17 30 93
3 11 19 39 40 48 160
11 16 18 21 29 48 143
1 7 20 21 30 35 114
2 14 22 32 34 42 146
10 25 39 43 47 48 212
9 15 30 31 40 47 172
19 22 24 29 34 43 171
1 17 19 20 22 32 111
3 7 17 31 33 45 136
2 9 12 21 36 41 121
3 12 27 30 31 33 136
2 13 16 26 27 31 115
4 8 16 31 33 39 131
5 12 29 31 33 39 149
7 10 19 28 29 30 123
2 12 25 35 46 47 167
1 11 23 30 35 42 142
5 12 17 31 32 46 143
4 7 11 13 17 23 75
11 14 24 30 35 41 155
2 8 13 30 34 43 130
10 18 22 23 25 40 138
2 6 8 17 30 34 97
1 2 19 26 34 45 127
16 21 24 35 39 48 183
7 8 23 25 28 47 138
3 4 6 23 35 38 109
3 6 23 29 36 45 142
6 19 20 29 30 31 135
16 25 35 45 47 48 216
6 10 12 20 21 37 106
9 14 18 23 41 47 152
10 12 19 38 42 47 168
5 6 7 30 31 39 118
1 6 10 19 22 34 92
13 19 30 35 42 43 182
16 27 34 40 44 45 206
6 19 27 37 38 41 168
12 13 19 21 27 42 134
7 9 12 16 24 43 111
4 6 9 15 27 47 108
6 8 13 26 29 38 120
12 20 25 27 32 41 157
6 15 23 25 39 43 151
“Winning” Lotto Strategies 5
9 12 31 42 46 48 188
16 18 26 37 40 43 180
5 6 17 21 23 38 110
4 7 18 30 37 47 143
5 9 18 21 23 47 123
5 6 13 39 42 45 150
5 18 22 23 30 40 138
3 6 14 18 24 47 112
5 16 20 26 44 47 158
1 2 17 21 40 41 122
1 19 27 28 45 47 167
1 7 10 34 37 43 132
13 15 16 19 39 48 150
6 11 14 15 22 25 93
4 5 10 24 28 33 104
4 25 26 30 35 48 168
3 5 6 9 15 38 76
2 13 15 17 19 20 86
4 10 19 30 32 38 133
7 12 14 31 43 46 153
23 25 28 35 41 47 199
1 18 19 27 34 38 137
7 17 18 28 36 42 148
2 3 8 15 23 46 97
5 7 8 9 23 38 90
10 11 16 27 28 36 128
24 30 33 39 41 47 214
1 3 12 15 21 23 75
14 25 28 42 43 48 200
5 13 15 16 22 28 99
7 12 19 20 25 40 123
1 3 17 28 39 44 132
5 7 15 18 32 40 117
7 13 21 26 29 34 130
9 14 16 18 21 38 116
1 25 26 28 39 48 167
4 14 27 32 37 48 162
1 3 5 16 26 28 79
8 10 17 25 32 42 134
4 8 15 23 25 42 117
2 15 39 40 44 47 187
11 20 23 34 37 46 171
5 12 19 24 26 41 127
10 12 24 31 32 33 142
11 18 22 30 32 42 155
4 16 17 19 23 45 124
8 14 19 23 38 41 143
7 8 11 28 33 45 132
2 8 10 20 21 25 86
10 19 23 34 40 44 170
2 3 20 22 33 42 122
4 12 16 24 31 39 126
15 16 20 28 31 39 149
1 3 8 13 33 35 93
12 19 31 37 42 43 184
13 20 22 28 33 38 154
9 12 14 15 26 27 103
20 21 24 28 29 35 157
21 22 30 33 35 41 182
12 23 28 35 41 47 186
11 20 26 29 40 43 169
9 10 13 30 32 36 130
8 17 20 26 36 38 145
11 21 24 27 43 46 172
1 5 33 34 41 42 156
8 9 21 31 43 48 160
3 5 13 26 31 45 123
5 6 18 39 41 47 156
3 5 14 24 35 48 129
22 36 43 45 46 47 239
12 17 20 22 23 44 138
1 8 14 27 30 32 112
2 9 30 39 43 47 170
2 14 20 22 33 42 133
7 19 32 35 37 45 175
4 20 25 27 34 43 153
6 18 29 30 36 37 156
1 3 6 16 23 35 84
3 7 15 26 35 45 131
3 13 15 21 25 44 121
1 2 7 29 45 46 130
1 4 19 20 45 47 136
5 7 9 17 20 39 97
3 20 24 31 32 39 149
17 18 29 34 41 43 182
17 24 29 42 44 47 203
5 8 14 20 21 34 102
4 9 25 27 31 47 143
3 4 20 23 26 38 114
8 23 26 27 32 36 152
3 19 32 33 41 44 172
4 9 22 23 27 28 113
17 20 25 39 42 47 190
1 4 10 14 22 38 89
2 6 22 23 31 34 118
9 19 23 30 37 43 161
5 21 22 24 29 40 141
3 12 32 33 36 41 157
5 20 30 31 40 41 167
3 4 6 14 17 40 84
We find that the mean of the sums of the winning numbers is 142.7. Then we
find that the median of this distribution is 143. Therefore, we know that the distribution
is close to symmetrical. Now, if you were using this system, you would select your
numbers so that their sum equaled 143. As you begin to do this you realize that there are
numerous ways to choose numbers that fit this criteria. By running a simple counter
program in Pascal* to add up all these possibilities, you will find that there are 147,670
such ways! Furthermore, upon observation we find that 143 has not appeared as the sum
of the winning numbers in the last 89 drawings. We note also that the number 143 has
only occurred as the sum five times in the last 273 drawings, and of those few
occurrences, you have a 1:147,670 per ticket chance of choosing the winning number. I
would hardly call this probability good.
The “Killer Lotto” Strategy
The author of this system, like many others, sells computer programs, which are
necessary for his strategy to work. His major claim is that “numbers show a bias towards
“Winning” Lotto Strategies 6
being drawn with the rest of the lotto numbers” (Saliu, 2002). His computer programs
sort out the numbers that appear as the most frequent pairs. The table below is an
example the author uses from a 6/69 game.
Number 1 Hits 19
With #: 8 16 30 22 25 14 31 32 42 27 29 17 21 15 33 35 36
Hits: 4 4 4 3 3 3 3 3 3 2 2 2 2 2 2 2 2
With #: 39 5 44 51 52 53 54 58 63 68 7 18 3 37 9 40 41
Hits: 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1
With #: 23 43 10 45 47 48 49 50 26 12 28 13 56 57 4 61 62
Hits: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
With #: 2 64 65 6 69 24 59 60 38 11 46 34 55 66 67 19 20
Hits: 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
Pairs total 95
This proposition of bias would be valid for any small finite set of drawings, which
is what the author uses to show his program works. He fails to mention how many
drawings this sample is taken from. Obviously, his data for the pairs of the number 1
came from only 19 samples. This does not give us an accurate assessment of the
occurrences of these pairs.
We can better understand the anticipated behavior of this experiment by using the
law of large numbers. Without loss of generality, we analyze all of the drawings where
the number 1 was selected, to show that over time, the probability diminishes of having a
number with a bias of being drawn with the number 1.
The law of large numbers, P[ |p_ - p| > _ ] ≤ (p_q)/(n_ _2), “declares that no
matter how small an _ is specified, the probability P that the sample probability (p_ )
differs from the single-trial probability of success (p) by more than _ can be made
arbitrarily small by sufficiently increasing the number of trials n” (Epstein, 1977, p. 28).
Now to apply the law of large numbers to our sample of lotto numbers that
include 1, we will let p = n_pi, where n is the number of trials and pi is the single-trial
probability of the number X being drawn. The sample (observed) probability of X being
drawn is p_. Now we let _ be arbitrarily small, and as we take more trials (increase n) we
see that (p_q) / (n_ _2) converges to zero. Thus, by our inequality we see that P(|p_ - p|)
also converges to zero; this says that the probability (P) that the sample probability (p_)
“Winning” Lotto Strategies 7
differs from the single-trial probability of success (p) approaches zero as the number of
trials increases. In other words, the probability (P) of the difference of the probability of
the number X being chosen when the number 1 is also chosen (p_) and the probability of
the single-trial probability (p), approaches zero as the number of games you play
increases. Thus, in the long run the bias will not be so significant, and this system will
not appear so amazing.
Systems
The systems that we have discussed thus far are based on past results from the
game. Lotto is a game of chance, based on a random selection of numbers. It is not
possible to consistently predict a random event. Consequently, do not concern yourself
with coupon patterns, “overdue” numbers, frequency statistics or other implausible good
luck charms. Although interesting, these systems have absolutely no bearing on the
ability to predict the winning numbers, nor the chances of winning a prize. A random
selection of numbers cannot, by definition, form a pattern.
Whether you play the four game minimum entry once a month, or invest $10,000
every week - every game played by either strategy has as much chance of winning a prize
as all the others. Although it seems unlikely, even the numbers - 1, 2, 3, 4, 5, 6 - have as
much chance of being drawn together as any other combination of six numbers.
The strategies that we will now discuss deal with the lotto in a more mathematical
approach and rely on past observations only for supportive evidence.
The “Delta Number” Strategy
Delta numbers are created by subtracting a number from the number following it.
For instance, take the lotto number 2 – 5 – 9 – 19 – 20 – 39,
it’s delta number would be: 2 – 3 – 4 – 10 – 1 – 19.
The idea of delta numbers comes from computers and the way that they store data
in memory. By compressing data in such a way, they are able to hold more data. In fact,
this idea of delta numbers and the lotto, emerged as the author of one article was working
on computer problems.
“Winning” Lotto Strategies 8
This particular author makes some surprising claims. First, he observes that the
delta number 1 appears 15% of the time. At first, this may seem like a good bet.
However, upon further evaluation you will find that the number of ways to "set up" that
delta number 1 varies greatly. Try it for yourself. In the boxes below, choose six delta
numbers where at least one of them being one. I can then change any of the surrounding
numbers (especially those preceding the 1’s) which will change your real lotto number.
Your Delta #
Your Lotto #
From a textbook on introductory combinatorics we learn that the number of
solutions to the equation x1 + x2 + x3 + x4 + x5 + x6 ≤ 48
such that 1 ≤ x1, x2, x3, x4, x5, x6 ≤ 43
can be found by simply by introducing yi = xi – 1 and letting S be the set of all non-
negative integral solutions of
y1 + y2 + y3 + y4 + y5 + y6 ≤ 42. Thus, the size of S, or the number of solutions of
S, is equal to ∑42 C((n+5),n) = 12,271,512 (Brualdi, 1999, p. 171). n=0
Therefore, there are 12,271,512 different ways to choose a delta number so that
the sum is less than or equal to 48. It might be a surprise to the author of this system that
the number of ways of choosing six numbers from a list of 48 is also 12,271,512.
Another way to think about this is that the relationship between lotto numbers and
their delta numbers is one-to-one, as illustrated with a bipartite graph. For every lotto
number, there is one and only one delta number, and for every delta number, there is one
and only one corresponding lotto number. Thus by using delta numbers, the lotto player
does not actually increase his/her odds of winning.
The Quick and Easy Way
The author of this lotto strategy preys upon the modern-day tendency of doing
things the quick and easy way. He professes to have discovered in his research a pattern
of winning numbers. He claims that "for the majority of all lotteries, the most numbers
are drawn from the sets of numbers with final digits 1-2-3-4-5-6" (Castor-Pollux
“Winning” Lotto Strategies 9
Publications, 2000). For example, take the Oregon Megabucks, which is a 6/48 game.
The table shows the numbers of the lotto, sorted in columns by the last digit.
1 2 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 23 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48
It is quite apparent that the numbers with final digits 1-2-3-4-5-6 make up 30 out
of 48 (62.5%) of the total numbers. Obviously then, the majority of the numbers drawn
in the lotto will come from this category. Rolling a die with the claim that it’s likely to
get a number between 1 and 6 isn’t much different. Just to show that this is true in the
lotto, when examining the last 50 drawings of the Oregon Megabucks these numbers
came up 62.58% of the time, just as they should.
The author also maintains that one may "wheel a selection of numbers from all
the numbers" (Castor-Pollux Publications, 2000) in this category and have a greater
chance of winning. Conveniently, the wheel and pattern can be found in the author’s
company magazine, one that he hopes you will subscribe to.
I assume that this wheel and system may be mathematically perfect, but would
only state the obvious and not give any advantage to winning the lotto.
Wheeling SystemsOne of the most popular lottery products sold is generally called the “wheel.”
Unlike the previous systems that we have analyzed, these systems do not claim to have
found a “trick” to guessing the lotto number that will win the jackpot. Instead, they
denounce such practices and take a different approach towards bringing “fortune.” We
will now examine a typical wheeling system.
In2play wheeling system “It does not matter how you pick your numbers - it's what you do with those numbers that
counts. This is the core element of in2play Lotto Systems. You can use any numbers you choose,they are designed specifically to win the most likely prizes, and are significantly more cost-effective than standard Systems Entries. And yes, you can win the 1st Division prize” (in2play,2003).
The purpose of the wheel is to give you as many possible combinations of your selectednumbers, so that you might have a better chance of winning at least a lesser prize. The
“Winning” Lotto Strategies 10
guarantee is “If there are SIX Winning numbers in your nine, then AT LEAST fivewill be in one line” (in2play, 2001).
The following is an example of an in2play wheel.
Selection 8 numbers
Games 7 (which is 25% of the standard Systems 8 Entry cost)
Guarantee Match four of the drawn numbers (at least three must be winning numbers) with your System selections, and you will win at leastone four-number prize (4th or 5th Division).
Instructions 1. Enter your selections in the top row of blank squares.2. Copy each number to every blank square in the column below it.3. Transfer each horizontal line of six-number combinations tomark a game on your lotto coupon.
In2play will be happy to sell you any one of their countless wheeling systems.
You can even go in on one with a friend. The downside to these systems is that they take
a hefty wallet both to purchase and to play. In2play advertises the obvious in that the
more you play, the more numbers you will cover, and thus the more likely you are to win.
These specific wheels sell anywhere from $15 to $178, (Order form). Then you must
figure in the amount of money that it will cost for you to play all of the number
combinations that the wheel instructs you to, for the 20 or 30 times you choose to play.
Assume that you use the above wheel to play the Oregon State Megabucks where
$1 gets you two tickets. (For our illustration and simplification, we will suppose that you
can buy one Megabucks ticket for $.50.) You play 7 lotto tickets on every night there is a
lotto drawing (since that is how many combinations our wheel gives), hoping to win at
least a match-3 prize, for which you would win $4. At this rate, you would have to win at
least a match-3 prize, seven out of every eight weeks just to cover your playing costs, not
“Winning” Lotto Strategies 11
to mention the cost of the wheel. As you can see, it doesn’t take long before you have
sunk a lot of money into the lotto.
To illustrate this point, I randomly chose eight numbers from 1-48 and then used
the Reduced ECONO system 8 wheel to create seven lotto tickets. I then examined the
data for Megabucks for the past 136 straight weeks as if I had played. Playing this many
games would have cost me $952.00 in tickets alone. I won 20 match-3 prizes, three
match-4 prizes, zero match-5 prizes and zero match-6 prizes. Since the match-4 and
match 5 prizes were pari-mutual, I had to find out the amount of each on each week that I
won. After adding it all up, I found I won a total of $227. That means that after 136
weeks of playing lotto, I’m $725 in the hole.
The bottom line is wheels do not give you better odds on winning the jackpot.
Their only advantage is that IF ALL the winning numbers are included in the set of
numbers that you have chosen, THEN you are guaranteed at least a match-3 or 4 or 5
prize, depending on the wheel. Remember that the more numbers the wheel allows you
to play increases your odds of guessing the winning six, but the cost of playing increases
dramatically so you must win more often to break even. Keep in mind that each ticket
you hold has just the same probability of being chosen as any other. However, if you are
bent on playing and plan to play more than one ticket with a certain set of “lucky”
numbers, wheels will help you create multiple tickets with the best possible
combinations.
Fairness
Bias
Of course, patterns observed in winning numbers open up the door for a few
questions. Is the method of selecting numbers fair (unbiased)? Many things may play a
role in the bias of a lotto. The numbered balls may not be exactly identical. If the
number seven ball is even slightly heavier than the others are, this will cause it to linger
around the bottom of the “cage” and not be sucked up by the vacuum tube, lowering its
probability of being selected. On the other hand, if the balls are drawn from the bottom
of the cage, its probability of selection would be greater than the other numbered balls.
“Winning” Lotto Strategies 12
The above scenarios of a biased lotto system are possible, but very unlikely. State
lotto organizations spend thousands of dollars to make sure that the lotto is unbiased. If it
were not so, people would catch on and the lotto would stand to lose millions of dollars.
To avoid this, balls are changed frequently, cages are maintained, and many different sets
of cages and balls are used to assure randomness. For example, in Oregon, “state
detectives oversee and are present at all drawings. The Oregon Lotto's random number
generator is tested and certified by an external lab and the Oregon State Police” (Oregon
Lotto, 2003).
Now we will show how fairness may be verified. To do this, let us look at the
Oregon State Lotto game, “Megabucks.”
Oregon MegabucksIn Oregon Megabucks, a player selects six numbers from a set of 48 possible
numbers (1-48). A player wins by matching 3, 4, 5, or 6 of the drawn numbers. Hence,
the odds of matching all six and winning the jackpot are 1:12,271,513.
In May of 2001, Megabucks made a significant changes to its layout. Changing
from a ball-selected machine to a random number generator to help ensure complete
randomness was the first change. The second was a change from the 6/44 number system
to a 6/48 system (Oregon Lotto, 2003). This almost cut in half the odds of winning the
jackpot from 1:7,059,053 to 1:12,271,513. Due to these significant changes, we will only
look at those drawings that have occurred since May 20, 2001.
Observations
Since May 21, 2001, there have been 273 drawings of six numbers each. That is
1638 winning numbers. Upon observation of the recent data of Megabucks, you may
notice right away that the numbers 14 and 23 have been drawn more than any other
number. This may not startle you until you look at the table of winning numbers and
realize that 14 has been chosen in eight of the last 25 drawings, including seven of the
last 12! The number 23, though not as impressive, still has remarkably appeared in five
of the last 25 drawings.
“Winning” Lotto Strategies 13
Most recent
Winning Numbers
7 14 18 29 30 32
3 6 14 22 26 28
3 6 14 24 26 28
12 16 24 29 34 46
2 18 23 30 31 46
8 15 22 28 32 46
8 24 26 28 37 45
1 14 17 18 26 43
1 14 25 30 38 44
2 5 27 34 40 45
11 14 18 36 41 42
2 5 9 14 34 47
3 15 31 44 45 48
9 15 17 18 22 23
5 20 31 32 36 47
5 9 18 24 38 39
1 7 26 29 37 45
18 19 34 37 38 42
3 5 23 29 36 38
2 6 23 38 43 46
4 16 18 21 40 45
20 22 28 37 42 46
1 2 16 27 33 34
14 15 25 38 39 42
3 4 23 31 42 44
6 10 15 26 42 47
6 9 12 18 24 40
8 12 13 15 30 43
12 18 24 27 32 47
6 13 17 20 25 27
13 14 17 31 35 43
9 15 19 21 29 44
2 7 15 29 37 42
11 21 27 28 36 38
20 23 24 27 32 35
1 4 5 14 27 31
4 18 24 34 37 47
15 25 29 35 38 47
12 14 18 20 36 37
15 20 25 38 42 43
8 14 22 26 27 29
4 11 14 32 38 45
11 21 43 44 46 47
7 12 21 33 38 39
5 15 17 38 40 46
5 11 17 20 26 46
10 14 21 29 33 35
4 8 9 13 23 29
1 27 30 34 36 40
7 13 14 17 25 42
4 7 14 15 40 45
11 19 24 29 37 39
5 12 16 23 34 38
4 10 17 18 24 26
1 2 5 24 27 47
8 12 17 24 27 48
8 13 26 34 44 48
31 35 36 37 40 42
9 12 24 30 31 47
10 20 38 43 44 47
3 11 18 26 39 48
8 10 12 27 39 42
6 20 25 38 39 41
1 10 11 15 42 45
11 13 23 35 43 46
13 14 18 24 41 48
2 17 32 33 43 44
1 12 26 34 38 44
3 15 16 28 33 37
1 5 13 24 40 46
5 7 9 21 34 47
1 13 28 33 46 48
10 15 24 35 39 43
4 15 28 37 44 45
10 31 37 43 44 47
1 14 16 24 25 33
16 22 23 24 27 33
10 22 25 32 44 46(Due to lack of space this is all the table we
will include here)
You’ll notice the frequency of the number 36 being drawn is relatively low (only
17 times). The numbers 46 and 48 don’t appear to be so hot either, that is, until you look
at the table above of recent winning numbers. You will notice that the number 46 has
appeared in five of the last 25 drawings. That’s as frequent as 23! So you might see why
0
5
1015
20
25
30
3540
45
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46
Frequency
“Winning” Lotto Strategies 14
it would be easy for someone to think they have found a winning pattern. However, by
applying the Chi-Square test we are able to determine if this game is unbiased.
Chi-Square Test
We apply the Chi-Square test
(where xi = the observed frequency of the ith number and Ei = the expected frequency of
the ith number) to the following table of drawn lotto numbers to see if the lotto is indeed,
unbiased.
c-square testNumber Number Expected Number Number Expected
of times number of times number
selected of times selected of times
1 38 34.125 25 34 34.125
2 34 34.125 26 38 34.125
3 38 34.125 27 39 34.125
4 34 34.125 28 30 34.125
5 39 34.125 29 32 34.125
6 41 34.125 30 35 34.125
7 31 34.125 31 39 34.125
8 32 34.125 32 31 34.125
9 32 34.125 33 28 34.125
10 30 34.125 34 33 34.125
11 27 34.125 35 32 34.125
12 43 34.125 36 17 34.125
13 37 34.125 37 29 34.125
14 45 34.125 38 39 34.125
15 42 34.125 39 33 34.125
16 32 34.125 40 32 34.125
17 33 34.125 41 29 34.125
18 37 34.125 42 35 34.125
19 33 34.125 43 35 34.125
20 38 34.125 44 30 34.125
21 36 34.125 45 32 34.125
22 30 34.125 46 26 34.125
“Winning” Lotto Strategies 15
23 44 34.125 47 41 34.125
24 37 34.125 48 26 34.125
Number of winning balls = 1638 Value of the c-square test = 0.793833445
By running the chi-square test on an excel spreadsheet, we find that the results of
the lotto thus far fall within 79.38% of the expected distribution. Thus, we can reject the
hypothesis that the Oregon Megabucks game is bias. Remember that this is only taking
into account the last 273 drawings.
Conclusion
Upon analysis of the Oregon State Lotto game, “Megabucks,” (which is a widely
used lotto game) we have shown that it is indeed, unbiased. The chi-square test may be
used to discover if other lotto games are also unbiased. Thus we may not be fooled into
buying every “winning lotto strategy” that we see because we may verify that the lotto
we are playing does not have tendencies or faults.
Consequently we see that there are several different types of “winning” lotto
strategies out there, each with their own devices and computer programs. I have only
examined a few here, but it should be noted that there are many more. Some make
erroneous claims due to ignorance or misunderstanding of statistics and mathematics,
while others make similar invalid assertions to try and sell their product. An analysis of
each individual strategy would be useful in identifying illogical claims.
* program Lotto;{Program to count (and display) all the possible ways a set of 6 distinctpositive integers can be chosen so that each integer is less than or equal to48 and the sum of the six is 143.}
uses Crt;
const yes = 1; no = 0; unintelligible = -1;
var Display, N1, N2, N3, N4, N5, N6, Pause, Sum : integer; Possibilities : Longint; AnswerKey : string; k : char;
begin ClrScr; Writeln('Possibility Counter'); Writeln; Writeln('Counts the possible ways a set of 6 distinct positive integers'); Writeln('less than or equal to 48 can be chosen so that their sum is 143.'); Writeln('(Note: it is assumed that the order these numbers are chosen is'); Writeln('unimportant.)');
“Winning” Lotto Strategies 16
Writeln; Writeln; Write('Do you wish to display the generated possibilities (Y/N)? '); Display:=unintelligible; While Display = unintelligible do begin Readln(AnswerKey); if ((AnswerKey = 'Y') or (AnswerKey= 'y')) then Display := yes else if ((AnswerKey='N') or (AnswerKey='n')) then Display := no else begin GotoXY(59,9); Write(' '); GotoXY(59,9); end; end; Writeln; Write('Do you wish to pause the display at the end of every screen (Y/N)? '); Pause:=unintelligible; While Pause = unintelligible do begin Readln(AnswerKey); if ((AnswerKey = 'Y') or (AnswerKey= 'y')) then Pause := yes else if ((AnswerKey='N') or (AnswerKey='n')) then Pause := no else begin GotoXY(68,11); Write(' '); GotoXY(68,11); end; end; Possibilities:=0; for N1:=1 to 43 do for N2:=N1+1 to 44 do for N3:=N2+1 to 45 do for N4:=N3+1 to 46 do for N5:=N4+1 to 47 do for N6:=N5+1 to 48 do begin Sum:=N1+N2+N3+N4+N5+N6; if Sum = 143 then begin Possibilities:=Possibilities+1; if Display = yes then writeln('#',Possibilities,': ',N1,' ',N2,' ',N3,' ',N4,' ',N5,' ',N6); if ((Pause = yes) and (WhereY = 24)) then begin Writeln('Press any key to continue'); k := ReadKey; ClrScr; end; end; end; Writeln; Writeln('There are ',Possibilities,' possible ways to pick six distinct integers'); Writeln('between 1 and 48 whose sum is 143.');end.