Analysis of 3-digit Lotery Games

Groupping and probabilities of 3-Digit Games

There are 1000 possible 3-digit numbers, and they are: 000, 001, 002, 003, 004, ... , 999. That is, just start from 0 and count upto 999. You can generate the full list of all 3-digit numbers by cliking here.

Each one of the 1000 numbers in 3-digit lottery games 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 mathamatical 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 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 123,123, 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-repeating numbers, one-repeating (doubles) numbers, and triples. Very often, non-repeating and one-repeating boxes are referred to as 6-way and 3-way boxes (or combos), respectively.

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-reapeating numbers. Similarly, there are 270/3=90 distinct one-repeating numbers. The following table summarizes the foregoing facts and the probability of winning.

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%). Analysis of actual winning numbers of all the 3-digit games in the United States continuously conducted by US-Lotteries.com reveal that the above figures very closely match the actual data.

Sum of Digits of 3-Digit Games

The sum of the 3 digits of the lottery games ranges from 0 to 27. The digits of the number 000 sum up to 0, and that of 999 sum up to 27, the rest lie in between. Well over 50% of 3-digit numbers have a sum between 10 and 17, it is therefore not surprizing to see winning numbers with sums 10-17 drawn every other day.

You might be tempted to play all the numbers with sums that range from 10 to 17 everyday in order to win every other day. This is quite naive since you are covering about 50% of the numbers and since the house edge of a lottery game is about 50%, you will only be breaking even every other day and losing in rest. However, it is not a bad idea to analise the trend of the sums in the past several days and play those that are overdue for a few days.

Click here to open a new window of the sums, odds and probabilities of all the 28 possible 3-digits and the corresponding distinct 3-digit numbers.