You may have read on the main game page that the chances of winning the Lucky for Life top prize are 1 in 30,821,472 or that you have an overall chance of 1 in 7.77 to win a prize on each ticket. But how exactly are those chances of winning a prize determined?
Lucky for Life is a 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 FIVE numbers out of a field of 48 (numbers 1 to 48) and then the second matrix, you select ONE number out of a field of 18 (numbers 1 to 18) – this number is called the Lucky Ball in this game. Like other split-matrix games, these two drawings are independent events played simultaneously. In order to figure the chances for winning, we must consider both events so the chances for each event are multiplied in order to determine the chances for winning one of the prize levels.
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 five number combinations can be made out of the field of 48 numbers which is then multiplied by the number of one digit combinations out of a field of 18. 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 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 5 numbers out of the set of 48 to the number of successful choices and number of failed choices. So your first number has a 1 in 48 chance of being drawn and then your second number has a 1 in 47 chance, and so on and can be written as “1/48 x 1/47 x 1/46 x 1/45 x 1/44” 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 all other smaller numbers to find probabilities. Let's use 5 as an example: 5! would be calculated as 5 x 4 x 3 x 2 x 1 = 120.
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.
To determine the overall chances in a game, all of the chances for each winning level are added together. The overall chances of winning Lucky for Life are 1 in 7.77. The formula for the overall chances is:
1 / ((1/30,821,472) + (1/1,813,027.76) + (1/143,355.68) + (1/8,432.69) + (1/3,413.23) + (1/200.78) + (1/249.75) + (1/14.69) + (1/49.95) + (1/32.02)) = 7.77