Everyone wants to get rich, especially if it only costs them $1.
Fortunately, many local state governments host lotteries, allowing
their constituents to donate cash into the budget in hopes of winning a
multi-million dollar prize. In practice, most lottery drawings consist
of a series of balls drawn randomly from a chamber which should
guarantee a fair opportunity for everyone to win. Theories of rigged
lotteries and fraud, however, run rampant across the Internet.1
The system is accused of not holding live drawings, publishing winning
numbers prior to drawing them, permitting the tweaking of data archives
to avoid payouts, intentionally modifying balls, or using balls with
painted numbers whose natural weight affects their likelihood of
appearance.

Rather than debunk any of these theories of lottery fraud or
rigging, this article reveals the trends and patterns of winning
lottery numbers for public scrutiny using basic data analysis. It uses
the results of the MegaMillions lottery and consists of the following
analyses:

• distribution of winning numbers over time
• behavioral stratification of numbers based on numerical position
• relationship between mutually winning numbers
• common differences between winning numbers
• winning number frequency

While such scrutiny has the potential to yield useful results, such
as identifying the existence or lack of "better numbers" to play, it is
presented so as to appeal to those interested in number patterns.

MegaMillions History

Recently, MegaMillions drew the largest jackpot ever recorded at $370 million, exceeding the previous record held by PowerBall.2
The prizes were not always so large, nor did the participants span the
United States. Beginning life in 1996, MegaMillions originally existed
under a different nomer: "The Big Game." For two years this lottery was
drawn weekly on Fridays until 1998, when a Tuesday drawing was added.
Over the past eleven years, the number of participating states has
doubled from only six to twelve. Although there are minor interstate
variations regarding how jackpots are paid to winners, the basic game
play remains the same.3

A single dollar in MegaMillions purchases a 1 in 175,711,536 chance
of landing the jackpot. A player may opt for a "QuickPick" set of
numbers generated automatically by a computer or they may choose to
select their own numbers. Since 2005, MegaMillions allows players to
choose five numbers between 1 and 56 plus a sixth number, the MegaBall,
between 1 and 46. This, however, was not always the selection pool.
When the "Big Game" was conceived, players were given a pool of numbers
1 through 50 to choose for their first five balls and numbers 1 through
25 for their sixth. Beginning in 1999, players were offered the numbers
1 through 50 for the five regular balls and 1 through 36 for the sixth.
When the game became MegaMillions in 2002, players selected numbers
between 1 and 52 for both the five regular balls and the MegaBall.4

Gathering Data

As a first step, it was necessary to obtain a collection of
MegaMillions' lottery numbers. Fortunately, the New Jersey Lottery
website has an archive of all winning numbers since September 6, 1996.5 As an added bonus, the archive of numbers exists in both HTML format for a pretty web presentation and
as a delimited file which is conducive for importing into a database.
For the purposes of this analysis, the winning lottery numbers were
imported into Microsoft SQL Server Express for processing queries.
Subsequent graphs were then created with Microsoft Excel to visualize
the trends and behavior.6

The delimited file of winning lottery numbers contained the results for 1078 drawings and provides the following fields:

• Year - formatted as YYYY
• Month - formatted as MM
• Day - formatted as DD
• Day of Week - formatted as Tuesday and Friday
• Ball 1, 2, 3, 4, 5 - as an integer
• MegaBall - as an integer
• Prize Payout - when present, formatted as a decimal value
• Date - formatted as YYYYMMDD

Distribution of Winning Numbers Over Time

The first trend analyzed was whether or not the numbers occur with
an even distribution. Balls 1, 2, 3, 4 and 5 were consolidated into a
single list to analyze their overall frequency of occurrence. Each
separate version of the lottery - two editions of BigGame and two
editions of MegaMillions - were analyzed independently to identify any
outlying activity. Subsequently, a similar grouping was performed to
determine the distribution of the MegaBall number. The following charts
detail the number of times each number was selected over the course of
the the entire span of MegaMillion's drawings.

Behavioral Stratification of Numbers Based on Numerical Position

After looking at the behavior of the numbers in aggregate, the
occurrence of numbers respective to their position was analyzed.
Unfortunately, the lottery does not store the numbers in the order they
were drawn. Rather, the data file saves the winning lottery numbers in
ascending order.7 As such, positional analysis focused on how the numbers are stratified within their given position.

It is important to recognize the four variations of the lottery's
number pool has an impact on the ratio of occurrence for each number.
As such, the data was broken into four sets titled (uncreatively)
version 1, version 2, version 3 and version 4. Winning numbers per
position were counted to determine the numbers that win most frequently
within each set. Then, an aggregate winning percentage was assigned by
combining the win ratio of each set multiplied by a time factor to
obtain the overall likelihood of a number to win. The time factor
represents the percentage share of drawings per version, which equates
to 15.95%, 32.37%, 30.05% and 21.61%, respective to MegaMillions
versions one (original) through four (current).

Each of the six graphs represent the top fifteen numbers per position:

• Green bars represent the current version of MegaMillions where
  players choose from numbers 1 through 56 and a MegaBall number of 1
  through 46.
• Blue bars represent the weighted aggregation of a number's winning
  percentage from all MegaMillions drawing variations since 1996.
• The red line represents a five variable polynomial trend line to
  the winning percentage of the current MegaMillions drawing pool.

Relationship Between Mutually Winning Numbers

Additionally, an analysis was performed to determine which numbers
"win together." After all, a player does not need to pick all six
numbers in order to win money from MegaMillions. Therefore, all
possible combinations of balls 1, 2, 3, 4 and 5 were formed to analyze
the occurrence of ball relationships.

Doubles

There are ten combinations of ball pairs: [1 2], [1 3], [1 4], [1
5], [2 3], [2 4], [2 5], [3 4], [3 5], & [4 5]. Using the
MegaMillions data, there are 10,780 possible pairs of which 1503 are
unique over the lottery's lifetime. In the current version of
MegaMillions, there are 2330 possible pairs of which 1202 are unique.
Pairs of numbers occur quite frequently; 1426 pairs have occurred
10,703 times throughout the lottery's lifetime compared with 692 pairs
that have appeared 1820 times in since the fourth version of
MegaMillions began.

The graph at right shows that in the current version of
MegaMillions, pairs of numbers win repeatedly quite often. Over the
lottery's lifetime, particular pairs have won very regularly.

Triples

There are ten combinations of ball triples: [1 2 3], [1 2 4], [1 2
5], [1 3 4], [1 3 5], [1 4 5], [2 3 4], [2 3 5], [2 4 5], & [3 4
5]. There are 10,780 possible triples of which 8675 are unique over the
lottery's lifetime. Within the past version of MegaMillions, there are
2330 possible triples of which 2245 are unique. 1789 sets of triplets
have repeated 3894 times in the lifetime of the lottery. Five sets of
triplets have occurred five times and two sets of triplets have
occurred six times. In the past year, however, only twenty-three sets
of three balls have repeated twice.

In the graph to the right, there have been many repeat winning
combinations of three numbers in the lifetime of the lottery, although
it is a less frequent phenomenon in the current version.

Quadruples

There are five combinations of ball quadruples: [1 2 3 4], [1 2 3
5], [1 2 4 5], [1 3 4 5], & [2 3 4 5]. There are 5390 possible
quadruplets of which 5328 are unique over the lottery's lifetime. Since
June 22, 2005 there are 1165 unique combinations of possible
quadruplets. Sixty-two sets of four numbers have repeated twice since
MegaMillions began. There have been zero sets of quadruplets winning
more than once in the current version of MegaMillions.

Quintuples

There is only one combination of ball quintuples: [1 2 3 4 5]. Only
one set of five numbers has ever repeated twice in the history of
MegaMillions: (11, 14, 18, 33, 4.

Common Differences Between Winning Numbers

A natural extension of analyzing number groups was to identify the
trends by which numbers differ from one another. For example, while
probability gives the numbers 20, 21, 22, 23 and 24 the same chance of
appearing as any other combination, is it likely? The numbers for the
most recent version of MegaMillions were scrutinized to determine if
there is a common difference between each ball.

Winning Number Frequency

Analyzing the distribution of numbers over time only provided half
the picture in terms of any given number's propensity towards winning.
Another aspect to consider was the temporal frequency by which a number
wins. For example, a number may have won on thirty occasions, but maybe
they were all two years ago. To study this behavior, the time delta
between each number's appearance was cataloged to establish statistics
for all numbers and for each number across the lifetime of the fourth
version of MegaMillions. Then, the analysis was repeated using only the
most recent six months of data to identify the cross section of numbers
that win frequently consistently and which numbers are just a current
flash in the pan.

Conclusions

Interesting as these trends may be, they will not assist in making
the odds of winning the MegaMillions lottery any better if the system
is truly fair and random. However, in the event there is some peculiar
factor skewing the ball selection such that any of these trends
continue, a player stands a mildly better chance of winning a partial
prize through the selection of weighted numbers.