Tri-State Heads or Tails Probability Calculations - Vermont Lottery

Tri-State/Vermont Triple Play Probability Explanation
How to Play Triple Play
The Tri-State Triple Play game is played in the following way. The player picks 5 numbers from 1 to 45
and the computer system generates two additional random sets of 5 numbers. When the player uses
"Easy Pick", then all three set of numbers will be randomly selected.
The Lottery draws 5 Winning numbers from 1 to 45.
There are two interrelated games that the player participates in and can win from:
• The Base Game – where each of the three sets of player’s numbers is matched to the Lottery’s
set of Winning numbers
• The Multi-Hand Match-up Game – where a combination of the three sets of player numbers is
used to match-up against the Lottery’s set of Winning numbers and where the same numbers in
different player sets of numbers can be used more than once. For example, if the player has
the following numbers: 1 st set is [1,2,3,4,5] , the 2 nd set is [1,3,8,12,30], and 3 rd set is:
[1,18,21,33,42] and the Winning Number is: [1,3,18,36,45], then the “1 “appears 3 times, the
number “3“ appears 2 times, the number “18” appears one time, and the player has a six
(3+2+1) multi-hand match.
The Base Game winning tiers and prizes for each set of numbers is:
Tier
TRIPLE PLAY BASE GAME
Matched
Numbers
Prize
1 5 of 5 of 45 Jackpot
2 4 of 5 of 45 $ 200
3 3 of 5 of 45 $ 10
The Multi-Hand Match-up Game winning tiers and prizes is:
Tier
TRIPLE PLAY MATCH-UP
Sum of
Matched Numbers
Prize
1 8,9,10,11,12, 13,14, or 15 $ 500
2 7 $ 200
3 6 $ 20
4 5 $ 7
5 4 $ 3
6 0 $ 2
Where Sum of Matched Numbers = 0 means that there is no matching numbers.
Probability Calculations Analysis
The probability calculations for both games are interrelated. To best understand how to compute the probabilities
one must break up the analysis into four stages:
1 - 10

1. Matching a single set of player numbers to Lottery’s Winning numbers.
2. Combining the three set of numbers.
3. Multi-Hand match-up.
4. Over-All winning with Base and Multi-Hand games.
Matching 5 of 45 on Single Set of Numbers
Given a set of five numbers of 45, then
Total Ways Single Set = COMBIN (45, 5) = 1,221,759 ways
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. For example, there is only 1 way to match 5 of 5 of 45 numbers. Therefore, the probability of a
match is 1 in 1,221,759 ways.
The general formula of Ways to match w numbers is:
Ways to Match {w of 5 of 45} = COMBIN (5, w)*COMBIN (45-5, 5-w)
The w variable can have only six values (5, 4, 3, 2, 1, and 0) which determines the number of possible tiers. For
the Base Game, three tiers are winning tiers and 3 are not.
The Reciprocal Probability or 1/Probability for each tier is:
1/Probability {w} = (Total Ways Single Set) divided by (Ways to Match {w of 5 of 45})
The following table describes the formulas and count of all the ways to win and lose on the Base game and their
corresponding reciprocal probability.
Base Game
Winning Tiers
Tier
Matches
(w)
Formula for Ways Ways Probability 1/Probability
1 5 (COMBIN(5,5) * COMBIN(45-5,5-5) 1 8.1849E-07 1,221,759.000
2 4 (COMBIN(5,4) * COMBIN(45-5,5-4) 200 1.6370E-04 6,108.795
3 3 (COMBIN(5,3) * COMBIN(45-5,5-3) 7,800 6.3842E-03 156.636
Total Ways and Overall Ways to Win: 8,001 6.5488E-03 152.701
The notation 6.5488E-03 is the floating point notation representing of the value “ .0065488 “ and the “E-xx”
represents the decimal point moving to the left xx positions. An exponential value of “E+xx” means the decimal
point moves to the right xx positions.
2 - 10

Tier
Matches
(w)
Base Game
Losing Tiers
Formula for Ways Ways Probability 1/Probability
4 2 (COMBIN(5,2) * COMBIN(45-5,5-2) 98,800 8.0867E-02 12.366
5 1 (COMBIN(5,1) * COMBIN(45-5,5-1) 456,950 3.7401E-01 2.674
6 0 (COMBIN(5,0) * COMBIN(45-5,5-0) 658,008 5.3857E-01 1.857
Total Ways to Lose: 1,213,758 9.9345E-01 1.007
Sum of the Ways: 1,221,759
Combining the three set of Numbers:
From the above Base Game Winning Tier Table, the overall 1/Probability of winning on 1 set is: 152.701. Each of
the three set of numbers are independent. Therefore, one can combine the probabilities to derive the overall
probability of winning in the base game in the following way:
Let:
Prob [Win] = Probability of an Overall Win, or (1/152.701) = 6.5488E-03
Prob [Lose] = Probability of not winning, 1-Prob [Win] = (1-1/152.701) = 9.9345E-01
The probability of not having any wins on all three is:
Prob [Lose On All Games] = Prob [Lose] * Prob [Lose]* Prob [Lose] = 9.8048E-01
The probability of having at least one win is:
Prob [Over All Win Base Game = 1 – Prob [Lose On All Games] = 1.9518E-02
The reciprocal probability is 1/ Prob [At least one win] = 1/1.9518E-02 = 5.1235E+01 = 51.235
Multi-Hand match-up Probability:
As described above, each of the player’s selection of 5 numbers can have 6 possibilities of matching against the
Lottery’s winning number. These possibilities are 5, 4,3,2,1, or 0 matches. Since there are 3 set of numbers,
there are 6*6*6, or 216 arrangements of matches. The total number of unique 3 set tickets is:
COMBIN (45, 5) ^3 = 1,823,713,616,578,948,479
Each set of five numbers is independent and probability of arrangement is the product of the probability of each
match set probability.
Let:
Prob [w1 of 5 of 45] = Probability of matching w1 numbers in Set 1
Prob [w2 of 5 of 45] = Probability of matching w2 numbers in Set 2
Prob [w3 of 5 of 45] = Probability of matching w3 numbers in Set 3
The probability of arrangement w1 and w2 and w3 is:
Prob [w1 of 5 of 45] x Prob [w2 of 5 of 45] x Prob [w3 of 5 of 45]
3 - 10

The following breakout table describes all 216 arrangements, by Order Number and their corresponding
probabilities. The column “Win Both Games” is when an arrangement can win on both the Base Game and the
Multi-Hand Game. These Flags “Y” are used later on in the discussion of the over-all win in both games.
Order
No.
Multi-Hand
Probability Break-out Table
w1 w2 w3 w1+w2+w3 Probability of each of the Arrangements
1 0 0 0 0 5.3857E-01 * 5.3857E-01 * 5.3857E-01 = 1.5622E-01
2 0 0 1 1 5.3857E-01 * 5.3857E-01 * 3.7401E-01 = 1.0849E-01
3 0 0 2 2 5.3857E-01 * 5.3857E-01 * 8.0867E-02 = 2.3456E-02
4 0 0 3 3 5.3857E-01 * 5.3857E-01 * 6.3842E-03 = 1.8518E-03
5 0 0 4 4 5.3857E-01 * 5.3857E-01 * 1.6370E-04 = 4.7483E-05 Y
6 0 0 5 5 5.3857E-01 * 5.3857E-01 * 8.1849E-07 = 2.3741E-07 Y
7 0 1 0 1 5.3857E-01 * 3.7401E-01 * 5.3857E-01 = 1.0849E-01
8 0 1 1 2 5.3857E-01 * 3.7401E-01 * 3.7401E-01 = 7.5338E-02
9 0 1 2 3 5.3857E-01 * 3.7401E-01 * 8.0867E-02 = 1.6289E-02
10 0 1 3 4 5.3857E-01 * 3.7401E-01 * 6.3842E-03 = 1.2860E-03 Y
11 0 1 4 5 5.3857E-01 * 3.7401E-01 * 1.6370E-04 = 3.2974E-05 Y
12 0 1 5 6 5.3857E-01 * 3.7401E-01 * 8.1849E-07 = 1.6487E-07 Y
13 0 2 0 2 5.3857E-01 * 8.0867E-02 * 5.3857E-01 = 2.3456E-02
14 0 2 1 3 5.3857E-01 * 8.0867E-02 * 3.7401E-01 = 1.6289E-02
15 0 2 2 4 5.3857E-01 * 8.0867E-02 * 8.0867E-02 = 3.5220E-03
16 0 2 3 5 5.3857E-01 * 8.0867E-02 * 6.3842E-03 = 2.7805E-04 Y
17 0 2 4 6 5.3857E-01 * 8.0867E-02 * 1.6370E-04 = 7.1295E-06 Y
18 0 2 5 7 5.3857E-01 * 8.0867E-02 * 8.1849E-07 = 3.5648E-08 Y
19 0 3 0 3 5.3857E-01 * 6.3842E-03 * 5.3857E-01 = 1.8518E-03
20 0 3 1 4 5.3857E-01 * 6.3842E-03 * 3.7401E-01 = 1.2860E-03 Y
21 0 3 2 5 5.3857E-01 * 6.3842E-03 * 8.0867E-02 = 2.7805E-04 Y
22 0 3 3 6 5.3857E-01 * 6.3842E-03 * 6.3842E-03 = 2.1951E-05 Y
23 0 3 4 7 5.3857E-01 * 6.3842E-03 * 1.6370E-04 = 5.6286E-07 Y
24 0 3 5 8 5.3857E-01 * 6.3842E-03 * 8.1849E-07 = 2.8143E-09 Y
25 0 4 0 4 5.3857E-01 * 1.6370E-04 * 5.3857E-01 = 4.7483E-05 Y
26 0 4 1 5 5.3857E-01 * 1.6370E-04 * 3.7401E-01 = 3.2974E-05 Y
27 0 4 2 6 5.3857E-01 * 1.6370E-04 * 8.0867E-02 = 7.1295E-06 Y
28 0 4 3 7 5.3857E-01 * 1.6370E-04 * 6.3842E-03 = 5.6286E-07 Y
29 0 4 4 8 5.3857E-01 * 1.6370E-04 * 1.6370E-04 = 1.4432E-08 Y
30 0 4 5 9 5.3857E-01 * 1.6370E-04 * 8.1849E-07 = 7.2161E-11 Y
31 0 5 0 5 5.3857E-01 * 8.1849E-07 * 5.3857E-01 = 2.3741E-07 Y
32 0 5 1 6 5.3857E-01 * 8.1849E-07 * 3.7401E-01 = 1.6487E-07 Y
33 0 5 2 7 5.3857E-01 * 8.1849E-07 * 8.0867E-02 = 3.5648E-08 Y
34 0 5 3 8 5.3857E-01 * 8.1849E-07 * 6.3842E-03 = 2.8143E-09 Y
35 0 5 4 9 5.3857E-01 * 8.1849E-07 * 1.6370E-04 = 7.2161E-11 Y
36 0 5 5 10 5.3857E-01 * 8.1849E-07 * 8.1849E-07 = 3.6081E-13 Y
37 1 0 0 1 3.7401E-01 * 5.3857E-01 * 5.3857E-01 = 1.0849E-01
38 1 0 1 2 3.7401E-01 * 5.3857E-01 * 3.7401E-01 = 7.5338E-02
39 1 0 2 3 3.7401E-01 * 5.3857E-01 * 8.0867E-02 = 1.6289E-02
40 1 0 3 4 3.7401E-01 * 5.3857E-01 * 6.3842E-03 = 1.2860E-03 Y
41 1 0 4 5 3.7401E-01 * 5.3857E-01 * 1.6370E-04 = 3.2974E-05 Y
42 1 0 5 6 3.7401E-01 * 5.3857E-01 * 8.1849E-07 = 1.6487E-07 Y
43 1 1 0 2 3.7401E-01 * 3.7401E-01 * 5.3857E-01 = 7.5338E-02
44 1 1 1 3 3.7401E-01 * 3.7401E-01 * 3.7401E-01 = 5.2318E-02
Win Both
Games
4 - 10

Order
No.
Multi-Hand
Probability Break-out Table
w1 w2 w3 w1+w2+w3 Probability of each of the Arrangements
45 1 1 2 4 3.7401E-01 * 3.7401E-01 * 8.0867E-02 = 1.1312E-02
46 1 1 3 5 3.7401E-01 * 3.7401E-01 * 6.3842E-03 = 8.9305E-04 Y
47 1 1 4 6 3.7401E-01 * 3.7401E-01 * 1.6370E-04 = 2.2899E-05 Y
48 1 1 5 7 3.7401E-01 * 3.7401E-01 * 8.1849E-07 = 1.1449E-07 Y
49 1 2 0 3 3.7401E-01 * 8.0867E-02 * 5.3857E-01 = 1.6289E-02
50 1 2 1 4 3.7401E-01 * 8.0867E-02 * 3.7401E-01 = 1.1312E-02
51 1 2 2 5 3.7401E-01 * 8.0867E-02 * 8.0867E-02 = 2.4458E-03
52 1 2 3 6 3.7401E-01 * 8.0867E-02 * 6.3842E-03 = 1.9309E-04 Y
53 1 2 4 7 3.7401E-01 * 8.0867E-02 * 1.6370E-04 = 4.9511E-06 Y
54 1 2 5 8 3.7401E-01 * 8.0867E-02 * 8.1849E-07 = 2.4755E-08 Y
55 1 3 0 4 3.7401E-01 * 6.3842E-03 * 5.3857E-01 = 1.2860E-03 Y
56 1 3 1 5 3.7401E-01 * 6.3842E-03 * 3.7401E-01 = 8.9305E-04 Y
57 1 3 2 6 3.7401E-01 * 6.3842E-03 * 8.0867E-02 = 1.9309E-04 Y
58 1 3 3 7 3.7401E-01 * 6.3842E-03 * 6.3842E-03 = 1.5244E-05 Y
59 1 3 4 8 3.7401E-01 * 6.3842E-03 * 1.6370E-04 = 3.9087E-07 Y
60 1 3 5 9 3.7401E-01 * 6.3842E-03 * 8.1849E-07 = 1.9544E-09 Y
61 1 4 0 5 3.7401E-01 * 1.6370E-04 * 5.3857E-01 = 3.2974E-05 Y
62 1 4 1 6 3.7401E-01 * 1.6370E-04 * 3.7401E-01 = 2.2899E-05 Y
63 1 4 2 7 3.7401E-01 * 1.6370E-04 * 8.0867E-02 = 4.9511E-06 Y
64 1 4 3 8 3.7401E-01 * 1.6370E-04 * 6.3842E-03 = 3.9087E-07 Y
65 1 4 4 9 3.7401E-01 * 1.6370E-04 * 1.6370E-04 = 1.0022E-08 Y
66 1 4 5 10 3.7401E-01 * 1.6370E-04 * 8.1849E-07 = 5.0112E-11 Y
67 1 5 0 6 3.7401E-01 * 8.1849E-07 * 5.3857E-01 = 1.6487E-07 Y
68 1 5 1 7 3.7401E-01 * 8.1849E-07 * 3.7401E-01 = 1.1449E-07 Y
69 1 5 2 8 3.7401E-01 * 8.1849E-07 * 8.0867E-02 = 2.4755E-08 Y
70 1 5 3 9 3.7401E-01 * 8.1849E-07 * 6.3842E-03 = 1.9544E-09 Y
71 1 5 4 10 3.7401E-01 * 8.1849E-07 * 1.6370E-04 = 5.0112E-11 Y
72 1 5 5 11 3.7401E-01 * 8.1849E-07 * 8.1849E-07 = 2.5056E-13 Y
73 2 0 0 2 8.0867E-02 * 5.3857E-01 * 5.3857E-01 = 2.3456E-02
74 2 0 1 3 8.0867E-02 * 5.3857E-01 * 3.7401E-01 = 1.6289E-02
75 2 0 2 4 8.0867E-02 * 5.3857E-01 * 8.0867E-02 = 3.5220E-03
76 2 0 3 5 8.0867E-02 * 5.3857E-01 * 6.3842E-03 = 2.7805E-04 Y
77 2 0 4 6 8.0867E-02 * 5.3857E-01 * 1.6370E-04 = 7.1295E-06 Y
78 2 0 5 7 8.0867E-02 * 5.3857E-01 * 8.1849E-07 = 3.5648E-08 Y
79 2 1 0 3 8.0867E-02 * 3.7401E-01 * 5.3857E-01 = 1.6289E-02
80 2 1 1 4 8.0867E-02 * 3.7401E-01 * 3.7401E-01 = 1.1312E-02
81 2 1 2 5 8.0867E-02 * 3.7401E-01 * 8.0867E-02 = 2.4458E-03
82 2 1 3 6 8.0867E-02 * 3.7401E-01 * 6.3842E-03 = 1.9309E-04 Y
83 2 1 4 7 8.0867E-02 * 3.7401E-01 * 1.6370E-04 = 4.9511E-06 Y
84 2 1 5 8 8.0867E-02 * 3.7401E-01 * 8.1849E-07 = 2.4755E-08 Y
85 2 2 0 4 8.0867E-02 * 8.0867E-02 * 5.3857E-01 = 3.5220E-03
86 2 2 1 5 8.0867E-02 * 8.0867E-02 * 3.7401E-01 = 2.4458E-03
87 2 2 2 6 8.0867E-02 * 8.0867E-02 * 8.0867E-02 = 5.2883E-04
88 2 2 3 7 8.0867E-02 * 8.0867E-02 * 6.3842E-03 = 4.1750E-05 Y
89 2 2 4 8 8.0867E-02 * 8.0867E-02 * 1.6370E-04 = 1.0705E-06 Y
90 2 2 5 9 8.0867E-02 * 8.0867E-02 * 8.1849E-07 = 5.3525E-09 Y
91 2 3 0 5 8.0867E-02 * 6.3842E-03 * 5.3857E-01 = 2.7805E-04 Y
92 2 3 1 6 8.0867E-02 * 6.3842E-03 * 3.7401E-01 = 1.9309E-04 Y
Win Both
Games
5 - 10

Order
No.
Multi-Hand
Probability Break-out Table
w1 w2 w3 w1+w2+w3 Probability of each of the Arrangements
93 2 3 2 7 8.0867E-02 * 6.3842E-03 * 8.0867E-02 = 4.1750E-05 Y
94 2 3 3 8 8.0867E-02 * 6.3842E-03 * 6.3842E-03 = 3.2960E-06 Y
95 2 3 4 9 8.0867E-02 * 6.3842E-03 * 1.6370E-04 = 8.4513E-08 Y
96 2 3 5 10 8.0867E-02 * 6.3842E-03 * 8.1849E-07 = 4.2257E-10 Y
97 2 4 0 6 8.0867E-02 * 1.6370E-04 * 5.3857E-01 = 7.1295E-06 Y
98 2 4 1 7 8.0867E-02 * 1.6370E-04 * 3.7401E-01 = 4.9511E-06 Y
99 2 4 2 8 8.0867E-02 * 1.6370E-04 * 8.0867E-02 = 1.0705E-06 Y
100 2 4 3 9 8.0867E-02 * 1.6370E-04 * 6.3842E-03 = 8.4513E-08 Y
101 2 4 4 10 8.0867E-02 * 1.6370E-04 * 1.6370E-04 = 2.1670E-09 Y
102 2 4 5 11 8.0867E-02 * 1.6370E-04 * 8.1849E-07 = 1.0835E-11 Y
103 2 5 0 7 8.0867E-02 * 8.1849E-07 * 5.3857E-01 = 3.5648E-08 Y
104 2 5 1 8 8.0867E-02 * 8.1849E-07 * 3.7401E-01 = 2.4755E-08 Y
105 2 5 2 9 8.0867E-02 * 8.1849E-07 * 8.0867E-02 = 5.3525E-09 Y
106 2 5 3 10 8.0867E-02 * 8.1849E-07 * 6.3842E-03 = 4.2257E-10 Y
107 2 5 4 11 8.0867E-02 * 8.1849E-07 * 1.6370E-04 = 1.0835E-11 Y
108 2 5 5 12 8.0867E-02 * 8.1849E-07 * 8.1849E-07 = 5.4175E-14 Y
109 3 0 0 3 6.3842E-03 * 5.3857E-01 * 5.3857E-01 = 1.8518E-03
110 3 0 1 4 6.3842E-03 * 5.3857E-01 * 3.7401E-01 = 1.2860E-03 Y
111 3 0 2 5 6.3842E-03 * 5.3857E-01 * 8.0867E-02 = 2.7805E-04 Y
112 3 0 3 6 6.3842E-03 * 5.3857E-01 * 6.3842E-03 = 2.1951E-05 Y
113 3 0 4 7 6.3842E-03 * 5.3857E-01 * 1.6370E-04 = 5.6286E-07 Y
114 3 0 5 8 6.3842E-03 * 5.3857E-01 * 8.1849E-07 = 2.8143E-09 Y
115 3 1 0 4 6.3842E-03 * 3.7401E-01 * 5.3857E-01 = 1.2860E-03 Y
116 3 1 1 5 6.3842E-03 * 3.7401E-01 * 3.7401E-01 = 8.9305E-04 Y
117 3 1 2 6 6.3842E-03 * 3.7401E-01 * 8.0867E-02 = 1.9309E-04 Y
118 3 1 3 7 6.3842E-03 * 3.7401E-01 * 6.3842E-03 = 1.5244E-05 Y
119 3 1 4 8 6.3842E-03 * 3.7401E-01 * 1.6370E-04 = 3.9087E-07 Y
120 3 1 5 9 6.3842E-03 * 3.7401E-01 * 8.1849E-07 = 1.9544E-09 Y
121 3 2 0 5 6.3842E-03 * 8.0867E-02 * 5.3857E-01 = 2.7805E-04 Y
122 3 2 1 6 6.3842E-03 * 8.0867E-02 * 3.7401E-01 = 1.9309E-04 Y
123 3 2 2 7 6.3842E-03 * 8.0867E-02 * 8.0867E-02 = 4.1750E-05 Y
124 3 2 3 8 6.3842E-03 * 8.0867E-02 * 6.3842E-03 = 3.2960E-06 Y
125 3 2 4 9 6.3842E-03 * 8.0867E-02 * 1.6370E-04 = 8.4513E-08 Y
126 3 2 5 10 6.3842E-03 * 8.0867E-02 * 8.1849E-07 = 4.2257E-10 Y
127 3 3 0 6 6.3842E-03 * 6.3842E-03 * 5.3857E-01 = 2.1951E-05 Y
128 3 3 1 7 6.3842E-03 * 6.3842E-03 * 3.7401E-01 = 1.5244E-05 Y
129 3 3 2 8 6.3842E-03 * 6.3842E-03 * 8.0867E-02 = 3.2960E-06 Y
130 3 3 3 9 6.3842E-03 * 6.3842E-03 * 6.3842E-03 = 2.6021E-07 Y
131 3 3 4 10 6.3842E-03 * 6.3842E-03 * 1.6370E-04 = 6.6721E-09 Y
132 3 3 5 11 6.3842E-03 * 6.3842E-03 * 8.1849E-07 = 3.3361E-11 Y
133 3 4 0 7 6.3842E-03 * 1.6370E-04 * 5.3857E-01 = 5.6286E-07 Y
134 3 4 1 8 6.3842E-03 * 1.6370E-04 * 3.7401E-01 = 3.9087E-07 Y
135 3 4 2 9 6.3842E-03 * 1.6370E-04 * 8.0867E-02 = 8.4513E-08 Y
136 3 4 3 10 6.3842E-03 * 1.6370E-04 * 6.3842E-03 = 6.6721E-09 Y
137 3 4 4 11 6.3842E-03 * 1.6370E-04 * 1.6370E-04 = 1.7108E-10 Y
138 3 4 5 12 6.3842E-03 * 1.6370E-04 * 8.1849E-07 = 8.5540E-13 Y
139 3 5 0 8 6.3842E-03 * 8.1849E-07 * 5.3857E-01 = 2.8143E-09 Y
140 3 5 1 9 6.3842E-03 * 8.1849E-07 * 3.7401E-01 = 1.9544E-09 Y
Win Both
Games
6 - 10

Order
No.
Multi-Hand
Probability Break-out Table
w1 w2 w3 w1+w2+w3 Probability of each of the Arrangements
141 3 5 2 10 6.3842E-03 * 8.1849E-07 * 8.0867E-02 = 4.2257E-10 Y
142 3 5 3 11 6.3842E-03 * 8.1849E-07 * 6.3842E-03 = 3.3361E-11 Y
143 3 5 4 12 6.3842E-03 * 8.1849E-07 * 1.6370E-04 = 8.5540E-13 Y
144 3 5 5 13 6.3842E-03 * 8.1849E-07 * 8.1849E-07 = 4.2770E-15 Y
145 4 0 0 4 1.6370E-04 * 5.3857E-01 * 5.3857E-01 = 4.7483E-05 Y
146 4 0 1 5 1.6370E-04 * 5.3857E-01 * 3.7401E-01 = 3.2974E-05 Y
147 4 0 2 6 1.6370E-04 * 5.3857E-01 * 8.0867E-02 = 7.1295E-06 Y
148 4 0 3 7 1.6370E-04 * 5.3857E-01 * 6.3842E-03 = 5.6286E-07 Y
149 4 0 4 8 1.6370E-04 * 5.3857E-01 * 1.6370E-04 = 1.4432E-08 Y
150 4 0 5 9 1.6370E-04 * 5.3857E-01 * 8.1849E-07 = 7.2161E-11 Y
151 4 1 0 5 1.6370E-04 * 3.7401E-01 * 5.3857E-01 = 3.2974E-05 Y
152 4 1 1 6 1.6370E-04 * 3.7401E-01 * 3.7401E-01 = 2.2899E-05 Y
153 4 1 2 7 1.6370E-04 * 3.7401E-01 * 8.0867E-02 = 4.9511E-06 Y
154 4 1 3 8 1.6370E-04 * 3.7401E-01 * 6.3842E-03 = 3.9087E-07 Y
155 4 1 4 9 1.6370E-04 * 3.7401E-01 * 1.6370E-04 = 1.0022E-08 Y
156 4 1 5 10 1.6370E-04 * 3.7401E-01 * 8.1849E-07 = 5.0112E-11 Y
157 4 2 0 6 1.6370E-04 * 8.0867E-02 * 5.3857E-01 = 7.1295E-06 Y
158 4 2 1 7 1.6370E-04 * 8.0867E-02 * 3.7401E-01 = 4.9511E-06 Y
159 4 2 2 8 1.6370E-04 * 8.0867E-02 * 8.0867E-02 = 1.0705E-06 Y
160 4 2 3 9 1.6370E-04 * 8.0867E-02 * 6.3842E-03 = 8.4513E-08 Y
161 4 2 4 10 1.6370E-04 * 8.0867E-02 * 1.6370E-04 = 2.1670E-09 Y
162 4 2 5 11 1.6370E-04 * 8.0867E-02 * 8.1849E-07 = 1.0835E-11 Y
163 4 3 0 7 1.6370E-04 * 6.3842E-03 * 5.3857E-01 = 5.6286E-07 Y
164 4 3 1 8 1.6370E-04 * 6.3842E-03 * 3.7401E-01 = 3.9087E-07 Y
165 4 3 2 9 1.6370E-04 * 6.3842E-03 * 8.0867E-02 = 8.4513E-08 Y
166 4 3 3 10 1.6370E-04 * 6.3842E-03 * 6.3842E-03 = 6.6721E-09 Y
167 4 3 4 11 1.6370E-04 * 6.3842E-03 * 1.6370E-04 = 1.7108E-10 Y
168 4 3 5 12 1.6370E-04 * 6.3842E-03 * 8.1849E-07 = 8.5540E-13 Y
169 4 4 0 8 1.6370E-04 * 1.6370E-04 * 5.3857E-01 = 1.4432E-08 Y
170 4 4 1 9 1.6370E-04 * 1.6370E-04 * 3.7401E-01 = 1.0022E-08 Y
171 4 4 2 10 1.6370E-04 * 1.6370E-04 * 8.0867E-02 = 2.1670E-09 Y
172 4 4 3 11 1.6370E-04 * 1.6370E-04 * 6.3842E-03 = 1.7108E-10 Y
173 4 4 4 12 1.6370E-04 * 1.6370E-04 * 1.6370E-04 = 4.3867E-12 Y
174 4 4 5 13 1.6370E-04 * 1.6370E-04 * 8.1849E-07 = 2.1933E-14 Y
175 4 5 0 9 1.6370E-04 * 8.1849E-07 * 5.3857E-01 = 7.2161E-11 Y
176 4 5 1 10 1.6370E-04 * 8.1849E-07 * 3.7401E-01 = 5.0112E-11 Y
177 4 5 2 11 1.6370E-04 * 8.1849E-07 * 8.0867E-02 = 1.0835E-11 Y
178 4 5 3 12 1.6370E-04 * 8.1849E-07 * 6.3842E-03 = 8.5540E-13 Y
179 4 5 4 13 1.6370E-04 * 8.1849E-07 * 1.6370E-04 = 2.1933E-14 Y
180 4 5 5 14 1.6370E-04 * 8.1849E-07 * 8.1849E-07 = 1.0967E-16 Y
181 5 0 0 5 8.1849E-07 * 5.3857E-01 * 5.3857E-01 = 2.3741E-07 Y
182 5 0 1 6 8.1849E-07 * 5.3857E-01 * 3.7401E-01 = 1.6487E-07 Y
183 5 0 2 7 8.1849E-07 * 5.3857E-01 * 8.0867E-02 = 3.5648E-08 Y
184 5 0 3 8 8.1849E-07 * 5.3857E-01 * 6.3842E-03 = 2.8143E-09 Y
185 5 0 4 9 8.1849E-07 * 5.3857E-01 * 1.6370E-04 = 7.2161E-11 Y
186 5 0 5 10 8.1849E-07 * 5.3857E-01 * 8.1849E-07 = 3.6081E-13 Y
187 5 1 0 6 8.1849E-07 * 3.7401E-01 * 5.3857E-01 = 1.6487E-07 Y
188 5 1 1 7 8.1849E-07 * 3.7401E-01 * 3.7401E-01 = 1.1449E-07 Y
Win Both
Games
7 - 10

Order
No.
Multi-Hand
Probability Break-out Table
w1 w2 w3 w1+w2+w3 Probability of each of the Arrangements
189 5 1 2 8 8.1849E-07 * 3.7401E-01 * 8.0867E-02 = 2.4755E-08 Y
190 5 1 3 9 8.1849E-07 * 3.7401E-01 * 6.3842E-03 = 1.9544E-09 Y
191 5 1 4 10 8.1849E-07 * 3.7401E-01 * 1.6370E-04 = 5.0112E-11 Y
192 5 1 5 11 8.1849E-07 * 3.7401E-01 * 8.1849E-07 = 2.5056E-13 Y
193 5 2 0 7 8.1849E-07 * 8.0867E-02 * 5.3857E-01 = 3.5648E-08 Y
194 5 2 1 8 8.1849E-07 * 8.0867E-02 * 3.7401E-01 = 2.4755E-08 Y
195 5 2 2 9 8.1849E-07 * 8.0867E-02 * 8.0867E-02 = 5.3525E-09 Y
196 5 2 3 10 8.1849E-07 * 8.0867E-02 * 6.3842E-03 = 4.2257E-10 Y
197 5 2 4 11 8.1849E-07 * 8.0867E-02 * 1.6370E-04 = 1.0835E-11 Y
198 5 2 5 12 8.1849E-07 * 8.0867E-02 * 8.1849E-07 = 5.4175E-14 Y
199 5 3 0 8 8.1849E-07 * 6.3842E-03 * 5.3857E-01 = 2.8143E-09 Y
200 5 3 1 9 8.1849E-07 * 6.3842E-03 * 3.7401E-01 = 1.9544E-09 Y
201 5 3 2 10 8.1849E-07 * 6.3842E-03 * 8.0867E-02 = 4.2257E-10 Y
202 5 3 3 11 8.1849E-07 * 6.3842E-03 * 6.3842E-03 = 3.3361E-11 Y
203 5 3 4 12 8.1849E-07 * 6.3842E-03 * 1.6370E-04 = 8.5540E-13 Y
204 5 3 5 13 8.1849E-07 * 6.3842E-03 * 8.1849E-07 = 4.2770E-15 Y
205 5 4 0 9 8.1849E-07 * 1.6370E-04 * 5.3857E-01 = 7.2161E-11 Y
206 5 4 1 10 8.1849E-07 * 1.6370E-04 * 3.7401E-01 = 5.0112E-11 Y
207 5 4 2 11 8.1849E-07 * 1.6370E-04 * 8.0867E-02 = 1.0835E-11 Y
208 5 4 3 12 8.1849E-07 * 1.6370E-04 * 6.3842E-03 = 8.5540E-13 Y
209 5 4 4 13 8.1849E-07 * 1.6370E-04 * 1.6370E-04 = 2.1933E-14 Y
210 5 4 5 14 8.1849E-07 * 1.6370E-04 * 8.1849E-07 = 1.0967E-16 Y
211 5 5 0 10 8.1849E-07 * 8.1849E-07 * 5.3857E-01 = 3.6081E-13 Y
212 5 5 1 11 8.1849E-07 * 8.1849E-07 * 3.7401E-01 = 2.5056E-13 Y
213 5 5 2 12 8.1849E-07 * 8.1849E-07 * 8.0867E-02 = 5.4175E-14 Y
214 5 5 3 13 8.1849E-07 * 8.1849E-07 * 6.3842E-03 = 4.2770E-15 Y
215 5 5 4 14 8.1849E-07 * 8.1849E-07 * 1.6370E-04 = 1.0967E-16 Y
216 5 5 5 15 8.1849E-07 * 8.1849E-07 * 8.1849E-07 = 5.4833E-19 Y
Total of All Probabilities = 1.0000000
Win Both
Games
To calculate the probability of each tier one must add all the probabilities for a given w1+w2+w3. For example,
the number of ways of matching a sum of 4 is 15
.
Order
No.
w1 w2 w3
Probability of Matching "w1+w2+w3 =4"
w1
+w2
+w3
Probability of each of the arrangements
5 0 0 4 4 5.3857E-01 * 5.3857E-01 * 1.6370E-04 = 4.7483E-05
10 0 1 3 4 5.3857E-01 * 3.7401E-01 * 6.3842E-03 = 1.2860E-03
15 0 2 2 4 5.3857E-01 * 8.0867E-02 * 8.0867E-02 = 3.5220E-03
20 0 3 1 4 5.3857E-01 * 6.3842E-03 * 3.7401E-01 = 1.2860E-03
25 0 4 0 4 5.3857E-01 * 1.6370E-04 * 5.3857E-01 = 4.7483E-05
40 1 0 3 4 3.7401E-01 * 5.3857E-01 * 6.3842E-03 = 1.2860E-03
45 1 1 2 4 3.7401E-01 * 3.7401E-01 * 8.0867E-02 = 1.1312E-02
50 1 2 1 4 3.7401E-01 * 8.0867E-02 * 3.7401E-01 = 1.1312E-02
8 - 10

Order
No.
w1 w2 w3
Probability of Matching "w1+w2+w3 =4"
w1
+w2
+w3
Probability of each of the arrangements
55 1 3 0 4 3.7401E-01 * 6.3842E-03 * 5.3857E-01 = 1.2860E-03
75 2 0 2 4 8.0867E-02 * 5.3857E-01 * 8.0867E-02 = 3.5220E-03
80 2 1 1 4 8.0867E-02 * 3.7401E-01 * 3.7401E-01 = 1.1312E-02
85 2 2 0 4 8.0867E-02 * 8.0867E-02 * 5.3857E-01 = 3.5220E-03
110 3 0 1 4 6.3842E-03 * 5.3857E-01 * 3.7401E-01 = 1.2860E-03
115 3 1 0 4 6.3842E-03 * 3.7401E-01 * 5.3857E-01 = 1.2860E-03
145 4 0 0 4 1.6370E-04 * 5.3857E-01 * 5.3857E-01 = 4.7483E-05
Sum of individual probabilities:
5.2360E-02
1/Probability of matching 4 19.0985
The full set of tiers is defined by the table below:
Multi-Hand Tier Probabilities
Tier
Sum
w1+w2+w3
Count of
Arrangements
Sum of
Probabilities
1/Probability
1 0 1 1.5622E-01 6.4012
2 1 3 3.2546E-01 3.0726
3 2 6 2.9638E-01 3.3740
4 3 10 1.5561E-01 6.4264
5 4 15 5.2360E-02 19.0985
6 5 21 1.1884E-02 84.1503
7 6 25 1.8657E-03 535.9933
8 7 27 2.0462E-04 4,887.0639
9 8 27 1.5654E-05
10 9 25 8.2558E-07
11 10 21 2.9354E-08
12 11 15 6.7908E-10
“ 8+” Lumped
13 12 10 9.6816E-12 60,572.5315
14 13 6 7.8631E-14
15 14 3 3.2900E-16
16 15 1 5.4833E-19
Totals: 216 1.0000E+00
The winning tiers are for Sum of Matches (w1+w2+w3) = 0,4,5,6,7,8,9,10,11,12,13,14 and 15. The 8+ is 8, 9, 10,
11,12,13,14 and 15 and lumped together. When the probabilities for these w1+w2+w3 are added, we get the
over-all probability of a match as 2.2255E-01 or 1/Probability of 4.4934.
9 - 10

Over All Probability of Winning:
The over-all probability of winning in either game or both is:
Prob [Over All Win Both Games] = + Prob [Over All Win Base Game]
+ Prob [Over All Win Multi-Hand Game]
- Prob [Over All Win Base Game and Over All Win Base Game]
The probability of winning in both games, the and condition, is determined from the Multi-Hand Break-out table
above. The value is derived by adding all the break-outs probabilities that correspond to the “Y” (Winner on both).
There are 186 break-outs of the 216 possible break-outs.
The calculation is:
And
Prob [Over All Win Base Game]
= + 1.9518E-02 (From Above)
Prob [Over All Win Multi-Hand Game]
= + 2.2255E-01 (From Above)
Prob [Over All Win Base Game and Multi-Hand Game] = - 1.3962E-02 (From Break-out Table)
Prob [Over All Win on Both Games]
= 2.2811E-01
1/Prob [Over All Win on Both Games]
= 4.3839E+00
Richard Mishelof
Games Design Consultants
Scientific Games International Corporation
Alpharetta, GA
June 18, 2005
10 - 10