Each one of the 1000 numbers in 3-digit 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 inexact 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 playingboxesandcombos(any order) of a 3-digit 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 non-repeating 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 3-digit numbers into three groups, namely,non-repeatingnumbers,one-repeating (doubles)numbers, andtriples. Very often, non-repeating and one-repeating boxes are referred to as6-wayand3-wayboxes (or combos), respectively. All the combinations of 3-digit games such as Cash 3, Play 3, Daily 3, and Pick 3 are listed below.
Of the 1000 3-digit numbers 720 are non-repeating, 270 one-repeating, and 10 triples. Since a non-repeating number has 5 more siblings, we can divide 720 by 6 to obtain 120 distinct non-repeating-digit numbers which are listed below.
3-digit, non-repeating combinations (6-way)
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 one-repeating numbers can be represented by one number, there are 270/3=90 distinct one-repeating numbers (doubles) as listed below.
3-digit, one-repeating combinations (Doubles) (3-way)
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 3-digit games.
| Group name and aliases | How many 3-dig 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 |
|---|---|---|---|---|
| Non-repeat (6-way) | 720 | 120 | 1:167 | 72% |
| One-repeat (3-way) (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 non-repeat numbers (72%), 27 one-repeats (27%), and 1 triple (1%). We have on this website analysis the actual winning numbers of all the 3-digit 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.
