Up to 10 possible template types can be obtained in this game.
3.3 Total Games by Type
Each template is formed by a given procedure and each type of template represents a single set of individual games. Based on the total number of games, it is revealed that not all games have the same probability to occur because two tens have nine numbers while three have ten numbers.
S (Sextet)
Each S template is formed by choosing six numbers from any ten.
2.C9,6 + 3.C10,6 = 168 + 630 = 798 games
Thus, in a draw of this lottery, the probability of a S-type game is
$$\frac{798}{\text{12,271,512}}\approx \text{6,5} . {10}^{-5}$$
3
The denominator 12,271,512 represents the total number of games of the lottery.
Qn (Quintet)
Each Qn template is formed by choosing five numbers from any ten and one number belonging to a different ten.
2.C9,5.C9,1 + 6.C9,5.C10,1 + 6.C10,5.C9,1 + 6.C10,5.C10,1 = 2268 + 7560 + 13608 + 15120 = 38556 games
Thus, in a draw of this lottery, the probability of a Qn-type game is
$$\frac{38556}{\text{12,271,512}}\approx \text{3,1} . {10}^{-3}$$
4
Q (Quartet)
Each Q template is formed by the choice of four numbers from any ten and two other numbers belonging to two distinct tens.
6.C9,4.C10,1.C10,1 + 6.C9,4.C9,1.C10,1 + 3.C10,4.C10,1.C10,1 + 3.C10,4.C9,1.C9,1 + 12.C10,4.C9,1.C10,1 = 75600 + 68040 + 63000 + 51030 + 226800 = 484470 games
Thus, in a random draw of this lottery, the probability of a Q-type game is
$$\frac{484470}{\text{12,271,512}}\approx \text{3,9} . {10}^{-2}$$
5
QP (Quartet-Pair)
Each QP template is formed by the choice of four numbers from any one ten and two other numbers belonging to another ten.
2.C9,4.C9,2 + 6.C9,4.C10,2 + 6.C10,4.C9,2 + 6.C10,4.C10,2 = 9072 + 34020 + 45360 + 56700 = 145152 games
Thus, in a random draw of this lottery, the probability of a QP-type game is
$$\frac{145152}{\text{12,271,512}}\approx \text{1,2} . {10}^{-2}$$
6
T(Trio)
Each T template is formed by the choice of three numbers from any one ten and three other numbers belonging to three other different tens.
2.C9,3.C10,1.C10,1.C10,1 + 6.C9,3.C9,1.C10,1.C10,1 + 6.C10,3.C9,1.C9,1.C10,1 + 6.C10,3.C10,1.C10,1.C9,1 = 168000 + 453600 + 583200 + 648000 = 1852800 games
Thus, in a random draw of this lottery, the probability of a T-type game is
$$\frac{1852800}{\text{12,271,512}}\approx \text{1,5} . {10}^{-1}$$
7
TT (Trio-Trio)
Each TT template is formed by the choice of three numbers from any one ten and three other numbers belonging to another ten.
1.C9,3.C9,3 + 6.C9,3.C10,3 + 3.C10,3.C10,3 = 7056 + 60480 + 43200 = 110736 games
Thus, in a random draw of this lottery, the probability of a TT-type game is
$$\frac{110736}{\text{12,271,512}}\approx \text{9,0} . {10}^{-3}$$
8
TP (Trio-Pair)
Each TP template is formed by the choice of three numbers from any ten, two numbers belonging to another ten, and one number belonging to another ten distinct from the previous.
6.C9,3.C9,2.C10,1 + 6.C9,3.C10,2.C9,1 + 6.C10,3.C9,2.C9,1 + 12.C9,3.C10,2.C10,1 + 12.C10,3.C9,2.C10,1 + 12.C10,3.C10,2.C9,1 + 6.C10,3.C10,2.C10,1 = 181440 + 204120 + 233280 + 453600 + 518400 + 583200 + 324000 = 2498040 games
Thus, in a random draw of this lottery, the probability of a TP-type game is
$$\frac{2498040}{\text{12,271,512}}\approx \text{2,0} . {10}^{-1}$$
9
P (Pair)
Each P template is formed by the choice of two numbers from any one ten and four other numbers belonging to other four different tens. Thus, we can form a total of
2.C9,2.C9,1.C10,1.C10,1.C10,1 + 3.C10,2.C9,1.C9,1.C10,1.C10,1 = 648000 + 1093500 = 1741500 games
Thus, in a random draw of this lottery, the probability of a P-type game is
$$\frac{1741500}{\text{12,271,512}}\approx \text{1,4} . {10}^{-1}$$
10
PP (Pair-Pair)
Each PP template is formed by the choice of two numbers from any one ten, two numbers belonging to another ten, and two other numbers belonging to two tens that are distinct from each other and the previous.
3.C9,2.C9,2.C10,1.C10,1 + 3.C10,2.C10,2.C9,1.C9,1 + 12.C9,2.C10,2.C9,1.C10,1 + 6.C9,2.C10,2.C10,1.C10,1 + 6.C10,2.C10,2.C9,1.C10,1 = 388800 + 492075 + 1749600 + 972000 + 1093500 = 4695975 games
Thus, in a random draw of this lottery, the probability of a PP-type game is
$$\frac{4695975}{\text{12,271,512}}\approx \text{3,8} . {10}^{-1}$$
11
PPP (Pair-Pair-Pair)
Each PPP template is formed by the choice of two numbers from any ten, two numbers belonging to another ten, and two numbers belonging to yet another ten that is distinct from the previous.
3.C9,2.C9,2.C10,2 + 6.C9,2.C10,2.C10,2 + 1.C10,2.C10,2.C10,2 = 174960 + 437400 + 91125 = 703485 games
Thus, in a random draw of this lottery, the probability of a PPP-type game is
$$\frac{703485}{\text{12,271,512}}\approx \text{5,7} . {10}^{-2}$$
12
Thus, we can organize 10 possible templates of the 6–48 lottery, as shown below in Table 21.
Table 21
Sampling distribution of the templates
Sampling distribution of the templates |
Template | # of template types | # of possible games | Probability |
S | 05 | 798 | 6.5.10− 5 = 0.0065% |
Qn | 20 | 38,556 | 3.1.10− 3 = 0.31% |
Q | 30 | 484,470 | 3.9.10− 2 = 3.9% |
QP | 20 | 145,152 | 1.2.10− 2 = 1.2% |
T | 20 | 1,852,800 | 1.5.10− 1 = 15% |
TT | 10 | 110,736 | 9.0.10− 3 = 0.9% |
TP | 60 | 2,498,040 | 2.0.10− 1 = 20% |
P | 5 | 1,741,500 | 1.4.10− 1 = 14% |
PP | 30 | 4,695,975 | 3.8.10− 1 = 38% |
PPP | 10 | 703,485 | 5.7.10− 2 = 5.7% |
TOTAL | 210 | 12,271,512 | 1 = 100% |