You may have read on the game page that the chances of winning the $250,000 top prize on Game 1 of $250K Triple Play are 1 in 487,635 or that you have an overall chance of 1 in 6.16 to win a prize on each ticket. But how exactly are those chances of winning a prize determined?
$250K Triple Play is basically a lotto game only with multiple sets of numbers playing for different prize levels. $250K Triple Play is played by selecting 4 numbers out of 60 possible choices (the numbers from 1 to 60).
So how does one determine what the chances of correctly matching all 4 numbers are for any of the three Games within $250K Triple Play, and how exactly the number of sets in each game is affected? The formula for figuring matching possibilities out of a set field of numbers is based on probability theory. It is basically the mathematical description of how many four number combinations can be made out of the field of 60 numbers.
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 4 numbers out of the set of 60 to the number of successful choices and number of failed choices. So your first number has a 1 in 60 chance of being drawn and then your second number has a 1 in 59 chance, and so on and can be written as “1/60 x 1/59 x 1/58 x 1/57” 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 figured 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 HYPGEOMDIST function that can calculate these chances easily. The equation below is how the math behind this function works.
So for Game 1 with one set of 4 numbers, you have a 1 in 487,635 chance of hitting all 4 of 4 and winning the $250,000 top prize. Hitting 4 of 4 for Games 2 and 3 it is a bit different since you have more sets of numbers. Each number set is independent from the other so you need to multiply the chances of each event to get the overall chances, and since the chances are the same for each set you are multiplying the chances times itself (i.e. exponentially) for however many number sets are at play.
Remember that when you see the chances as “1 in x” it means “1 divided by” so the formula for Game 2 matching 4 of 4 with three sets of numbers is:
Start with 4 of 4 formula above and then calculate this at the end: 1/(1-(1-1/487,635)^3) = 1 in 162,545.33
The formula for Game 3 matching 4 of 4 with five sets of numbers is:
Start with 4 of 4 formula above and then calculate this at the end: 1/(1-(1-1/487,635)^5) = 1 in 97,527.4
The formula is the same for 3 of 4 only substituting in the correct values for the successful and failed picks. So for one set, your chances would be 1 in 2,177 but you can win this for matching any of the 9 sets of numbers so you put it into the formula above to get the overall chances:
1/(1-(1-1/2,177)^9) = 1 in 242.33
The formula is the same for 2 of 4 only substituting in the correct values for the successful and failed picks. So for one set, your chances would be 1 in 52.77 but you can win this for matching any of the 9 sets of numbers so you put it into the formula above to get the overall chances:
1/(1-(1-1/52.77)^9) = 1 in 6.32
The overall chances per ticket are 1 in 6.16. The formula for the overall chances is:
1 / (1/487,635 + 1/162,545.33 + 1/97,527.4 + 1/242.33 + 1/6.32) = 6.16
The overall chances drop to 1 in 5.72 if you include the possibility of winning more than one prize on a ticket.