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United States Lotteries

Analysis of 4-digit Lotery Games

The analysis of 4-digit games involves the analysis of individual digits, pairs, sums, the numbers themselves in exact order (straight), numbers in any order (boxes), and so on. The purpose of 4-digit game analysis is to determine the most and least frequently drawn numbers, digits, pairs, etc., and to identify late entities in order to predict the winning numbers. Such analysis calls for the actual statistics of the drawn number which are available at different web sites. On this web site we shall see only the prominent features that lead to a more detailed analysis. Here is what's available on this page:

  • Groupping and probabilities of 4-Digit Games
  • Sum of Digits of 4-Digit Games
  • Groupping and probabilities of 4-Digit Games

    Each one of the 10000 numbers in 4-digit lottery games have equal chance of being drawn at any given time, therefore, the chance of winning with one selected number in exact order (straight) is 1 in 10,000. In mathamatical terms, we say the probability of winning is 0.0001 or 0.01%. On the other hand, when playing boxes and combos (any order) of a 4-digit game, the order in which the number is drawn doesn't matter, therefore instead of 10,000 possibilities we have only 715. In this later case, the numbers 1122,2211, 1221, 2112, 1212, 2121, for example, can be represented by only one number - 1122. For this reason, the chance of winning with a box or a combo play is obviously higher than the chance of winning with a straight play.

    Since there are 715 possible outcomes, one may be tempted to conclude that the chance winning in a box is 1 in 715. This is not exactly true since not all boxes have equal chances of winning; numbers with non-repeating digits have a better chance of winning than numbers with one digit repeated, two digits repeated, and one digit tripled. For example, the number 1234 has a better chance of winning than the number 1122, since 1234 covers 24 possible permutations (thus the name 24-way box), while 1122 covers only the six shown above.

    Consequently, we shall classify all the 4-digit numbers into five groups, namely, non-repeating numbers, doubles (one-repeating) numbers, dual-doubles (two-repeating), triples (one-tripled) and quadruplets (all four identical). Very often, the first four groups are referred to as 24-way, 12-way, 6-way, and 4-way boxes (or combos), respectively.

    Of the 10,000 4-digit numbers 5040 are non-repeating, 4320 one-repeating (doubles), 270 two-repeating (dual-doubles), 360 one-tripled, and 10 quadruples. Since a non-repeating number has 23 more siblings, we can divide 5040 by 24 to obtain 210 distinct non-reapeating numbers. Similarly, there are 4320/12=360 distinct one-repeating numbers, 270/6=45 distinct dual-doubles, 360/4=90 distinct triples.

    The following table and figures summarize the foregoing facts and the probability of winning.

    Group name and aliasesHow many 4-dig Numbers fall into this groupHow many distinct members of the groupodds of winning with a number in the groupProbability of a number drawn to be in the goup
    Non-repeating (24-way)50402101:41750.40%
    One-repeating (12-way) (Doubles)43203601:83343.20%
    Two-repeating (6-way) (Dual-Doubles)270451:16672.70%
    One-tripled (4-way) (triples)360901:25003.60%
    Quadruples10101:10,0000.1%

    4-digit probabilities       4-digit groups

    For instance, if you play 1234, 1122, and 0000, the chance of winning with these numbers in any arrangement are 1 in 417, 1 in 1,667, and 1 in 10,000, respectively.

    On the other hand, the probabilities listed above indicate that out of 1000 draws, we should theoretically expect 504 non-repeat numbers (50.40%), 432 doubles (43.20%), 27 dual-doubles (2.70%), 36 triples (3.6%), and 1 quadruple (0.1%). Analysis of actual winning numbers of all the 3-digit games in the United States continuously conducted by US-Lotteries.com reveal that the above figures very closely match the actual data.

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    Sum of Digits of 4-Digit Games

    The sum of the 4 digits of the lottery games ranges from 0 to 36. The digits of the number 0000 sum up to 0, and that of 9999 sum up to 36, the rest lie in between. Well over 50% of 4-digit numbers have a sum between 14 and 22, it is therefore not surprizing to see winning numbers with sums 14-22 drawn every other day.

    You might be tempted to play all the numbers with sums that range from 14 to 22 everyday in order to win every other day. This is quite naive since you are covering about 50% of the numbers and since the house edge of a state lottery game is 50%, you will only be breaking even every other day and losing in rest. However, it is not a bad idea to analise the trend of the sums in the past several days and play those that are overdue for a few days.

    Click here to open a new window of the sums, odds and probabilities of all the 37 possible 4-digits and the corresponding distinct 4-digit numbers.


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