Understanding MO Millions Chances

You may have read on the main game page that the chances of winning the MO Millions top prize are 1 in 5,245,786, or that you have an overall chance of 1 in 11.74 to win a prize for each play. But how exactly are those chances of winning a prize determined?

MO Millions is a modified split-matrix type lotto game, similar in game set-up as other split-matrix games, such as Powerball or Mega Millions. Split-matrix actually means there are two matrices, rather than just one, as in a simple lotto game. In the first matrix, you select SIX numbers out of a field of 42 (numbers 1 to 42). The second matrix is the Bulls-Eye number selection, comprised of the SIX winning numbers drawn in the first matrix. Like other split-matrix games, these two drawings are independent events. In order to figure the chances for winning, we must consider both events so that the chances for each event are multiplied in order to determine the chances for winning one of the prize levels. It should be noted that if you successfully pick the 6 winning numbers, you will also win the Bulls-Eye portion, too.

The formula for figuring matching possibilities out of a set field of numbers is based on probability theory. It is the mathematical description of how many six-number combinations can be made out of the field of 42 numbers, which is then multiplied by the chances out of the field of the 6 winning numbers. It is important to remember that the formula has to account for the numbers not drawn, as well as the numbers drawn. The formula used for this calculation is virtually the same one used for most lotto-type jackpot games. It uses sampling without replacement so that when a number is drawn, the next sample size is one smaller — you don't replace the number. There is a relationship of the total number of ways to select 6 numbers out of the set of 42 to the number of successful choices and number of failed choices. So, your first number has a 1 in 42 chance of being drawn, then your second number has a 1 in 41 chance, and so on. This can be written as "1/42 x 1/41 x 1/40 x 1/39 x 1/38 x 1/37" which can be expressed as a "binomial coefficient." This is also a combination function. The basic formula for a binomial coefficient uses what mathematicians call "factorials" and are depicted as x!, or a number followed by an exclamation point. A factorial is a number multiplied progressively by descending numbers to find probabilities. Let's use 6 as an example: 6! would be calculated as 6 x 5 x 4 x 3 x 2 x 1 = 720.

This type of formula is referred to as a "hypergeometric distribution". If you have Microsoft Excel, there is a HYPGEOM.DIST function that can calculate these chances easily. The equations below show how the math behind this function works. 

Overall Chances:
To determine the overall chances in a game, all of the chances for each winning level are added together. The overall chances of winning MO Millions are 1 in 11.73. The formula for the overall chances is:

1 / ((1/5,245,786) + (1/29,143.26) + (1/145,716.28) + (1/832.66) + (1/1,665.33) + (1/73.47) + (1/73.47) + (1/17.81)) = 11.73