Grouping and probabilities of 3Digit US Lottery Games
Each one of the 1000 numbers in 3digit lottery games (Cash 3, Play 3, Daily 3, Pick 3, The Numbers, Daily Numbers, Numbers, The Daily Game, Lucky Numbers)
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 1000.
In mathematical terms, we say the probability of winning is 0.001 or 0.1%. On the other hand, when playing boxes and combos (any order) of a 3digit game,
the order in which the number is drawn doesn't matter; therefore instead of 1000 possibilities we have only 220. In this later case, the numbers
123, 132, 213, 231, 312, 321, for example, are represented by only one number 123. 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 220 possible outcomes, one may be tempted to conclude that the chance winning in a box is 1 in 220.
This is not exactly true since not all boxes have equal chances of winning; numbers with nonrepeating digits have a
better chance of winning than numbers with one digit repeated. For example, the number 123 has a better chance of winning than
the number 122, since 123 covers six possible permutations shown above, while 122 covers only three (122, 212, 221).
Consequently, we shall classify all the 3digit numbers into three groups, namely, nonrepeating numbers,
onerepeating (doubles) numbers, and triples. Very often, nonrepeating and onerepeating boxes are referred to
as 6way and 3way boxes (or combos), respectively. All the combinations of 3digit games such as Cash 3, Play 3, Daily 3, and Pick 3
are listed below.
Of the 1000 3digit numbers 720 are nonrepeating, 270 onerepeating, and 10 triples. Since a nonrepeating number has 5 more siblings, we
can divide 720 by 6 to obtain 120 distinct nonrepeatingdigit numbers which are listed below.
3digit, nonrepeating combinations (6way)
012, 013, 014, 015, 016, 017, 018, 019, 023, 024, 025, 026, 027, 028, 029, 034, 035, 036, 037, 038, 039, 045, 046, 047, 048, 049, 056, 057, 058, 059, 067, 068, 069, 078, 079, 089, 123, 124, 125, 126, 127, 128, 129, 134, 135, 136, 137, 138, 139, 145, 146, 147, 148, 149, 156, 157, 158, 159, 167, 168, 169, 178, 179, 189, 234, 235, 236, 237, 238, 239, 245, 246, 247, 248, 249, 256, 257, 258, 259, 267, 268, 269, 278, 279, 289, 345, 346, 347, 348, 349, 356, 357, 358, 359, 367, 368, 369, 378, 379, 389, 456, 457, 458, 459, 467, 468, 469, 478, 479, 489, 567, 568, 569, 578, 579, 589, 678, 679, 689, 789
Similarly, since three onerepeating numbers can be represented by one number, there are 270/3=90 distinct
onerepeating numbers (doubles) as listed below.
3digit, onerepeating combinations (Doubles) (3way)
001, 002, 003, 004, 005, 006, 007, 008, 009, 011, 022, 033, 044, 055, 066, 077, 088, 099, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 334, 335, 336, 337, 338, 339, 344, 355, 366, 377, 388, 399, 445, 446, 447, 448, 449, 455, 466, 477, 488, 499, 556, 557, 558, 559, 566, 577, 588, 599, 667, 668, 669, 677, 688, 699, 778, 779, 788, 799, 889, 899
The following table and figures summarize the foregoing facts and the probability of winning with any order arrangement of 3digit games.
Group name and aliases  How many 3dig Numbers fall into this group  How many distinct members of the group  odds of winning with a number in the group  Probability of a number drawn to be in the group 
Nonrepeat (6way)  720  120  1:167  72% 
Onerepeat (3way) (Doubles)  270  90  1:333  27% 
Triples  10  10  1:1000  1% 
For instance, if you play 123, 112, and 000, the chance of winning with these numbers in any arrangement are 1 in 167, 1 in 333, and 1 in 1000, respectively.
On the other hand, the probabilities listed above indicate that out of 100 draws, we should theoretically expect 72 nonrepeat numbers (72%),
27 onerepeats (27%), and 1 triple (1%). We have on this website analysis the actual winning numbers of all the 3digit games of all states.
These analysis results show that the above figures very closely match the actual data. You can go to the "boxes(combos) analysis" of the Pick 3, Cash 3, Play 3, etc. pages of your state lotteries to see
the agreement between the theoretical expectations and the actual data.
Read more:
• US 3digit lottery games
• How to play 3digit games
• Sum of 3digit numbers
• 3digit combinations and permutations generator

