Tri-State Megabucks Probability Calculations

Tri-State Megabucks Probability Calculations
How to play Tri-State Megabucks
The player picks 6 numbers from 1 through 42, or chooses "Easy Pick" to select the six numbers.
On Wednesdays and Saturdays, 7 numbers (the 7th number is a Bonus number) are drawn from
a pool of 42 numbers by the lottery.
The winning Tiers are:
MEGA-BUCKS
Prize Tiers
Tier Match Prize
1 6 Numbers Jackpot
2 5 Numbers + Bonus $ 10,000
3 5 Numbers $ 1,000
4 4 Numbers + Bonus $ 50
5 4 Numbers $ 40
6 3 Numbers + Bonus $ 5
7 2 Numbers + Bonus $ 2
8 3 Numbers $ 1
Probability Calculations
Megabucks Game Probability
The players select 6 of 42 and the Lottery selects 6 primary numbers and one bonus number from a set of
42 numbers. The total number of ways that player can select combinations of six numbers is:
Total Ways = COMBIN(42,6) = 5,245,786
Using the Microsoft EXCEL notation, the function COMBIN(n,k) is the number of combinations of n items
taken k at a time. This formula is sometimes called: Combination formula or the Binomial formula. The
COMBIN function is defined as:
COMBIN(n,k) = n!/k!(n-k)!
Where the notation m! is called m factorial and is equal to m*(m-1)*(m-2)*(m-3)*...*2*1 and by
definition 0!= 1.
Probability is defined as the number of ways a match can be made divided by the Total Ways they can be
distributed.
The player’s selected six numbers when matched to Lottery’s selected 7 numbers can fall into three groups
of numbers:
1. Six primary numbers
2. One bonus number
3. Numbers not selected

There are 13 possible tier conditions (5,4,3,2,1,0 primary matches and 2 conditions on the bonus (match, no
match), or the condition of matching all six primary numbers matching; that is 6x2+1=13. Of the 13
conditions 8 are winning tiers and 5 are no prize tiers.
The combination formula for counting the number of ways to match the tiers is:
Ways {h hits and a bonus} = COMBIN(6,h)*COMBIN(1,a)*COMBIN(42-6-1,6-h-a)
Where:
h is the player matching the number of primary numbers (6,5,4,3,2,1,0)
a is the player matching the number of bonus number (1,0)
if h=6, then a=0
The Reciprocal Probability or 1/Probability for each tier is:
1/Probability{ w and a} = Total Ways/ Ways{ w and a }
The following table describes the formulas and count of all the ways to win and lose and their
corresponding reciprocal probability
Megabucks Game Probabilities
Winning Prize Tiers
Tier Match Formula Ways 1/Probability
1 6 Numbers (COMBIN(6,6) * COMBIN(1,0)) * (COMBIN(42-6-1,6-6-0) 1 5,245,786.000
2 5 Numbers + Bonus (COMBIN(6,5) * COMBIN(1,1)) * (COMBIN(42-6-1,6-5-1) 6 874,297.667
3 5 Numbers (COMBIN(6,5) * COMBIN(1,0)) * (COMBIN(42-6-1,6-5-0) 210 24,979.933
4 4 Numbers + Bonus (COMBIN(6,4) * COMBIN(1,1)) * (COMBIN(42-6-1,6-4-1) 525 9,991.973
5 4 Numbers (COMBIN(6,4) * COMBIN(1,0)) * (COMBIN(42-6-1,6-4-0) 8,925 587.763
6 3 Numbers + Bonus (COMBIN(6,3) * COMBIN(1,1)) * (COMBIN(42-6-1,6-3-1) 11,900 440.822
7 2 Numbers + Bonus (COMBIN(6,2) * COMBIN(1,1)) * (COMBIN(42-6-1,6-2-1) 98,175 53.433
8 3 Numbers (COMBIN(6,3) * COMBIN(1,0)) * (COMBIN(42-6-1,6-3-0) 130,900 40.075
Total Ways and Overall Ways to Win: 250,642 20.929
No Prize Tiers
Tier Match Formula Ways 1/Probability
9 2 Numbers (COMBIN(6,2) * COMBIN(1,0)) * (COMBIN(42-6-1,6-2-0) 785,400 6.679
10 1 Numbers + Bonus (COMBIN(6,1) * COMBIN(1,1)) * (COMBIN(42-6-1,6-1-1) 314,160 16.698
11 1 Numbers (COMBIN(6,1) * COMBIN(1,0)) * (COMBIN(42-6-1,6-1-0) 1,947,792 2.693
12 0 Numbers + Bonus (COMBIN(6,0) * COMBIN(1,1)) * (COMBIN(42-6-1,6-0-1) 324,632 16.159
13 No Matches (COMBIN(6,0) * COMBIN(1,0)) * (COMBIN(42-6-1,6-0-0) 1,623,160 3.232
Total Ways to Lose: 4,995,144 1.050
Total Ways: 5,245,786 1.000
Richard Mishelof
Games Design Consultants
Scientific Games International Corporation
Alpharetta, GA
March 31, 2004