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Grouping and probabilities of 4-Digit US Lottery GamesEach one of the 10,000 numbers in 4-digit lottery games (Cash 4, Play 4, Daily 4, Pick 4, DC-4, 4 Thisway,Win 4, Big 4, The 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 10,000. In mathematical 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-repeating-digit numbers which are listed below. 4-digit, non-repeating-digit combinations (24-way)
0123, 0124, 0125, 0126, 0127, 0128, 0129, 0134, 0135, 0136, 0137, 0138, 0139, 0145, 0146, 0147, 0148, 0149, 0156, 0157, 0158, 0159, 0167, 0168, 0169, 0178, 0179, 0189, 0234, 0235, 0236, 0237, 0238, 0239, 0245, 0246, 0247, 0248, 0249, 0256, 0257, 0258, 0259, 0267, 0268,
0269, 0278, 0279, 0289, 0345, 0346, 0347, 0348, 0349, 0356, 0357, 0358, 0359, 0367, 0368, 0369, 0378, 0379, 0389, 0456, 0457, 0458, 0459, 0467, 0468, 0469, 0478, 0479, 0489, 0567, 0568, 0569, 0578, 0579, 0589, 0678, 0679, 0689, 0789, 1234, 1235, 1236, 1237, 1238, 1239,
1245, 1246, 1247, 1248, 1249, 1256, 1257, 1258, 1259, 1267, 1268, 1269, 1278, 1279, 1289, 1345, 1346, 1347, 1348, 1349, 1356, 1357, 1358, 1359, 1367, 1368, 1369, 1378, 1379, 1389, 1456, 1457, 1458, 1459, 1467, 1468, 1469, 1478, 1479, 1489, 1567, 1568, 1569, 1578, 1579,
1589, 1678, 1679, 1689, 1789, 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, 2389, 2456, 2457, 2458, 2459, 2467, 2468, 2469, 2478, 2479, 2489, 2567, 2568, 2569, 2578, 2579, 2589, 2678, 2679, 2689, 2789, 3456, 3457, 3458, 3459, 3467,
3468, 3469, 3478, 3479, 3489, 3567, 3568, 3569, 3578, 3579, 3589, 3678, 3679, 3689, 3789, 4567, 4568, 4569, 4578, 4579, 4589, 4678, 4679, 4689, 4789, 5678, 5679, 5689, 5789, 6789
Similarly, since twelve one-repeating numbers can be represented by one number, there are 4320/12=360 distinct one-repeating-digit numbers (doubles) as listed below. 4-digit, one-repeating-digit (doubles) combinations (12-way)
0012, 0013, 0014, 0015, 0016, 0017, 0018, 0019, 0023, 0024, 0025, 0026, 0027, 0028, 0029, 0034, 0035, 0036, 0037, 0038, 0039, 0045, 0046, 0047, 0048, 0049, 0056, 0057, 0058, 0059, 0067, 0068, 0069, 0078, 0079, 0089, 0112, 0113, 0114, 0115, 0116, 0117, 0118, 0119, 0122, 0133, 0144, 0155, 0166, 0177, 0188, 0199, 0223, 0224, 0225, 0226, 0227, 0228, 0229, 0233, 0244, 0255, 0266, 0277, 0288, 0299, 0334, 0335, 0336, 0337, 0338, 0339, 0344, 0355, 0366, 0377, 0388, 0399, 0445, 0446, 0447, 0448, 0449, 0455, 0466, 0477, 0488, 0499, 0556, 0557, 0558, 0559, 0566, 0577, 0588, 0599, 0667, 0668, 0669, 0677, 0688, 0699, 0778, 0779, 0788, 0799, 0889, 0899, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1134, 1135, 1136, 1137, 1138, 1139, 1145, 1146, 1147, 1148, 1149, 1156, 1157, 1158, 1159, 1167, 1168, 1169, 1178, 1179, 1189, 1223, 1224, 1225, 1226, 1227, 1228, 1229, 1233, 1244, 1255, 1266, 1277, 1288, 1299, 1334, 1335, 1336, 1337, 1338, 1339, 1344, 1355, 1366, 1377, 1388, 1399, 1445, 1446, 1447, 1448, 1449, 1455, 1466, 1477, 1488, 1499, 1556, 1557, 1558, 1559, 1566, 1577, 1588, 1599, 1667, 1668, 1669, 1677, 1688, 1699, 1778, 1779, 1788, 1799, 1889, 1899, 2234, 2235, 2236, 2237, 2238, 2239, 2245, 2246, 2247, 2248, 2249, 2256, 2257, 2258, 2259, 2267, 2268, 2269, 2278, 2279, 2289, 2334, 2335, 2336, 2337, 2338, 2339, 2344, 2355, 2366, 2377, 2388, 2399, 2445, 2446, 2447, 2448, 2449, 2455, 2466, 2477, 2488, 2499, 2556, 2557, 2558, 2559, 2566, 2577, 2588, 2599, 2667, 2668, 2669, 2677, 2688, 2699, 2778, 2779, 2788, 2799, 2889, 2899, 3345, 3346, 3347, 3348, 3349, 3356, 3357, 3358, 3359, 3367, 3368, 3369, 3378, 3379, 3389, 3445, 3446, 3447, 3448, 3449, 3455, 3466, 3477, 3488, 3499, 3556, 3557, 3558, 3559, 3566, 3577, 3588, 3599, 3667, 3668, 3669, 3677, 3688, 3699, 3778, 3779, 3788, 3799, 3889, 3899, 4456, 4457, 4458, 4459, 4467, 4468, 4469, 4478, 4479, 4489, 4556, 4557, 4558, 4559, 4566, 4577, 4588, 4599, 4667, 4668, 4669, 4677, 4688, 4699, 4778, 4779, 4788, 4799, 4889, 4899, 5567, 5568, 5569, 5578, 5579, 5589, 5667, 5668, 5669, 5677, 5688, 5699, 5778, 5779, 5788, 5799, 5889, 5899, 6678, 6679, 6689, 6778, 6779, 6788, 6799, 6889, 6899, 7789, 7889, 7899
Again, since six dual-doubles numbers can be represented by one number, there are 270/6=45 distinct dual-double-digit numbers as listed below. 4-digit, dual-double-digit combinations (6-way)
0011, 0022, 0033, 0044, 0055, 0066, 0077, 0088, 0099, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3344, 3355, 3366, 3377, 3388, 3399, 4455, 4466, 4477, 4488, 4499, 5566, 5577, 5588, 5599, 6677, 6688, 6699, 7788, 7799, 8899
Finally, since four dual-doubles numbers can be represented by one number, there are 360/4=90 distinct one-tripled-digit numbers as listed below. 4-digit, one-tripled-digit combinations (4-way)
0001, 0002, 0003, 0004, 0005, 0006, 0007, 0008, 0009, 0111, 0222, 0333, 0444, 0555, 0666, 0777, 0888, 0999, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1222, 1333, 1444, 1555, 1666, 1777, 1888, 1999, 2223, 2224, 2225, 2226, 2227, 2228, 2229, 2333, 2444, 2555, 2666, 2777, 2888, 2999, 3334, 3335, 3336, 3337, 3338, 3339, 3444, 3555, 3666, 3777, 3888, 3999, 4445, 4446, 4447, 4448, 4449, 4555, 4666, 4777, 4888, 4999, 5556, 5557, 5558, 5559, 5666, 5777, 5888, 5999, 6667, 6668, 6669, 6777, 6888, 6999, 7778, 7779, 7888, 7999, 8889, 8999
The following table and figures summarize the foregoing facts and the probability of winning with 4-digit boxes.
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%). We have on this website analysis of the actual winning numbers of all the 4-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 4, Cash 4, Play 4, etc. pages of your state lotteries to see the agreement between the theoretical expectations and the actual data. Read more: • US 4-digit lottery games • How to play 4-digit games • Sum of 4-digit numbers • 4-digit combinations and permutations generator |
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