Grouping, combinations, and probabilities of 3-digit lottery games; how often is each group of numbers drawn and what are the odds.
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Pick 3 Lottery Games - Combinations and Probabilities

By : Melkam Dirset

Although each drawing of a Pick 3 lottery game does not have anything to do with the previous draws, there are still some patterns which are both theoretically and practically demonstrated to be true that every regular Pick 3 lottery player should be aware of them. In this article we shall see the frequency and probability of boxes (that is, any arrangement of the digits) of the Pick 3 lottery game by first classifying the numbers into three groups.

Note that even though we are using the name Pick 3 in this article, these lottery games are known by different names in different states such as Cash 3 in Florida, Tennessee, and Georgia; Daily 3 in California, Indiana, Michigan, Minnesota, and West Virginia; Play 3 in Connecticut and Delaware ; simply Numbers in New York, Pennsylvania, and the District Of Columbia; and The Daily Game in Washington. In fact, while a more appropriate and technical name would be 3-digit lottery games, we will use it here sparingly lest being too academic.

First of all, there are 1000 possible 3-digit numbers ranging from 000 to 999 and the chance of winning a Pick 3 game in exact order (also called straight) is therefore 1 in 1000. However, when dealing with boxes, where the order of the digits doesn't matter, there are only 220 possible combinations, resulting in a higher chance of winning, or lower odds. The chance of winning with a box is not nevertheless 1 in 220 since the boxes could be with or without repeating digits. For example, the number 360 has a better chance of winning than the number 007; and both have better chances than 555. The reason is that 360, having none of its digits repeated, covers six possible permutations (360, 306, 036, 630, 603, 063), while 007 covers only three (007, 700, 070), and 555 is only itself.

Consequently, Pick 3 numbers are classified into three groups, namely, Non-repeating, Doubles (or One-repeating), and Triples. Very often, the first two groups are referred to as 6-way and 3-way boxes (or combos), respectively. The 220 Pick 3 box combinations are composed of 120 non-repeating, 90 doubles and 10 triples. Now, the question is how often is a number drawn from each group and what are the probabilities of winning with a given 3-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 3 numbers fall into each group.

Out of the one thousand 3-digit numbers 720 are non-repeating, 270 are doubles, and 10 are triples. This means that for every 100 draws of a Pick 3 game, we should theoretically expect 72 non-repeating numbers (72%), 27 doubles (27%), and 1 triple (1%). A logical question at this point would be if the actual winning numbers of United States 3-digit lottery games demonstrate such a distribution; and indeed they do! If we look at the the Maryland Lottery, one of the oldest Pick 3 playing states, in the last 100 draws of the writing of this article (July 2009) there have been 70 non-repeating numbers, 29 doubles and 1 triple. In fact, although these figures are in line with the theoretical expectations, probability theories conform to real situations when the samples are large. Thus, if we analyze all the 16017 Pick 3 winning numbers of Maryland from 1976 to date, 11530 were non-repeating, 4334 doubles and 153 triples. These translate to 71.99%, 27.05%, and 0.96%, respectively. Such agreement of the theoretical expectations and the actual data are manifested not only in Maryland Pick 3, but also in all the 38 states that play 3-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.

Now, lets's see the chances of winning with a Pick 3 box play. If you bet on the boxes of a non-repeating number such as 360 and a double number such as 007, the odds of winning are about 1 in 167 for the first case, and 1 in 333 for the later. This is because 360, as seen above, wins with six numbers, thus 1000/6 = 166.7; while 007 wins with only three numbers, thus 1000/3 = 333.3. This explains why the pay-out for a winning double number is twice as much as a winning non-repeating number. Obviously, since there is no box play for triples, the pay-out is similar to a straight play. Most states pay $80 for $1 of non-repeating boxes, $160 for $1 of doubles, and $500 for $1 of triples. A word of caution: a common mistake is to consider the permutations of 077 as part of 007 boxes. The box of 077 is a completely different box than 007; in the first case the 7 is repeated, while in the later, it is the 0 that is repeated.

In conclusion, although it is hard to predict precisely a winning number, one of the things that a Pick 3 player should pay attention is to the trend of the patterns of the numbers in recent draws with respect to being non-repeating, doubles, or triples. 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 profit from Pick 3 games.


Author's Link: Latest and past results, statistics, analysis, and combinations generators of all United States 3-digit lottery games (Pick 3, Daily 3, Play 3, Cash 3, Numbers Game) are available at US-Lotteries.com.


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