Pick 4 Lottery Games - Combinations and Probabilities

Needless to say that each drawing of a Pick 4 lottery game is independent of the previous draws, however, every regular Pick 4 lottery player should be aware of some patterns which are both theoretically and practically demonstrated to be true. In this article we shall see the frequency and probability of boxes (that is, any arrangement of the digits) of the Pick 4 lottery game by first classifying the numbers into five groups.

Although we are using the name Pick 4 in this article, these lottery games are known by different names in different states. They are called Play 4 in Connecticut, Delaware, and Florida; Daily 4 in California, Indiana, Michigan, and Virginia; Cash 4 in Georgia and Tennessee. Other names include DC-4, Numbers, Win 4, and Big 4. In fact, while a more appropriate and technical name would be 4-digit lottery games, we will use it here sparingly lest being too academic.

There are 10,000 possible 4-digit numbers ranging from 0000 to 9999 and the chance of winning a Pick 4 lottery game in exact order (also called straight) is therefore 1 in ten thousand. However, when dealing with boxes, where the order of the digits doesn't matter, there are only 715 possible combinations resulting in a higher chance of winning or lower odds. The chance of winning with a box is not nevertheless 1 in 715 since the boxes could be with or without repeating digits. For example, the number 2013 has a better chance of winning than the numbers 2009, 2002, 2000, and 2222. The reason is that 2013, having none of its digits repeated, covers twenty-four possible permutations (0123, 0132, 0312, 3012, 0213, 0231, 0321, 3021, 2013, 2031, 2301, 3201, 1023, 1032, 1302, 3102, 1203, 1230, 1320, 3120, 2103, 2130, 2310, 3210), 2009 has one digit repeated and covers 12 numbers (0029, 0092, 0902, 9002, 0209, 0290, 0920, 9020, 2009, 2090, 2900, 9200), 2002 with two digits repeated covers 6 (0022, 0202, 2002, 0220, 2020, 2200), 2000 with one digit tripled covers 4 (0002, 0020, 0200, 2000), while 2222 is only itself.

Consequently, Pick 4 numbers are classified into five groups, namely, Non-repeating, Doubles (or One-repeating), Dual-doubles, Triples, and Quadruples. Very often, the first four groups are referred to as 24-way, 12-way, 6-way and 4-way boxes (or combos), respectively. The 715 Pick 4 box combinations are composed of:

210 Non-repeating,
360 Doubles,
45 Dual-doubles,
90 Triples, and
10 Quadruples.

The list of all Pick 4 combinations for each group can be found at our web site.

Now, the question is how often is a number drawn from each group and what are the probabilities of winning with a given 4-digit number when the order of the digits is not taken into consideration. In order to answer these questions, we have first to determine how many Pick 4 numbers fall into each group.

Out of the ten thousand 4-digit numbers, 5040 are Non-repeating, 4320 are Doubles, 270 Dual-doubles, 360 Triples, and 10 Quadruples. This means that for every 1000 draws of a Pick 4 game, we should theoretically expect 504 Non-repeating numbers (or 50.4 percent), 432 Doubles (43.2%), 27 Dual-doubles (2.7%), 36 Triples (3.6%), and 1 Quadruple (0.1%). A logical question at this point would be if the actual United States 4-digit lottery games demonstrate such a distribution; the answer is, indeed they do!

If we look at the the Illinois Lottery, one of the oldest Pick 4 playing states, in the last 1000 draws of the writing of this article (September 2009) there have been 506 Non-repeating numbers, 428 Doubles, 29 Dual-doubles, 32 Triples, and 5 Quadruples. In fact, although these figures are more or less in line with the theoretical expectations, probability theories conform to real situations when the samples are large. Thus, if we analyze all the 14183 Pick 4 draws of Illinois from 1983 to date, 7154 were Non-repeating, 6153 Doubles, 379 Dual-doubles, 486 Triples, and 11 Quadruples. These translate to 50.44%, 43.38%, 2.67%, 3.43%, and 0.08 percent, respectively. Such agreement of the theoretical expectations and the actual data are manifested not only in Illinois Pick 4, but also in all the 31 states that play 4-digit games as can be evidenced by the Boxes Analysis part of our web site. Certainly, since the data are dynamic, changing from day to day, even twice a day in many state lotteries, one should expect an occasional deviation from what is expected from probability.

What about the chances of winning with a Pick 4 box play? The odds of winning are calculated by dividing 10000 by the number of permutations of the group. For instance, for a Non-repeating number, there are 24 permutations, therefore 10000/24 = 416.7. It follows that the odds of winning with a Non-repeating number are 1 in 416.7. The pay-outs are also proportional to the odds. In fact, while you should theoretically be paid $416.70 for each dollar, the relentlessly exploitive state lotteries pay you only about $200; this is less than half of what it is worth. Some states may pay slightly more others may pay less. The odds and the common pay-outs for the five Pick 4 groups are listed below:

Non-repeating : 1 in 416.7, pays $200/$1
Doubles : 1 in 833.3, pays $400/$1
Dual-doubles : 1 in 1666.7, pays $800/$1
Triples : 1 in 2500, pays $1200/$1
Quadruples : 1 in 10000, pays $5000/$1

Although it is hard to predict precisely a winning number, it is important that a Pick 4 player pay attention to the trend of the patterns of the numbers in recent draws with respect to being Non-repeating, Doubles, Dual-doubles, Triples, or Quadruples. If one of these groups is too late, especially much beyond the expected time lapse, then it is advisable to concentrate on that group. More importantly, if coupled with other analyses such as frequency and lateness of digits, pairs, and the boxes themselves, one may successfully take advantage of Pick 4 games.