聰明組合的問題可以這樣理解:
假設:
n = 對某一系統的所有號碼 (例如:香港六合彩 49)
k = 每張票要填的號碼數目 (例如:香港六合彩 6)
p = 假設你選的一堆球數量而又真的開了 (例如: 6)
t = 最低中獎保證 (例如: 3)
例如: L(n,k,p,t) = L(49,6,3,5) 解讀為 由 49個球 選 6 個字而 假設你能估中 5 球保證最少中 3 的注數。為簡化我們會寫 3if5 (估中 5 球保證最少中 3 球)。這是最低保證，你亦有機會中 4,5 甚至 6 個字。
如果不用聰明組合,要在 49個球選6個的組合 (L49,6,6,6)是 49C6=13,983,816!
如果要選中1個字，我們有:
L(49,6,6,1)=Ceiling(49-6+1/6)
所以一般來說:
L(n,k,p,t)=Ceiling(n-p+1/k)
以香港六合彩為例，最少要中 3 個字而你又諗都唔諗的話你要:
87 <= L(49,6,6,3)<=163.
即是說，你買晒 163 注，必會中 1 注 最少 3 個字。 當然成本要 163*$10 = $1630.
Ticket 1 : 1 6 7 17 23 46
Ticket 2 : 1 6 9 25 32 45
Ticket 3 : 1 6 11 13 22 38
Ticket 4 : 1 6 20 27 34 47
Ticket 5 : 1 6 33 36 40 49
Ticket 6 : 1 7 9 22 34 49
...
Ticket 162 : 22 23 33 38 46 49
Ticket 163 : 25 32 33 34 47 49
但這 163 注不算分怖平衝，用 minimal overlapping algorithm 得出的結果更好。我在 第66期 (4/6/2011) 就查過，新的排列中最少 3注以上。
Gail Howard 本 Lottery Master Guide 有寫每種選擇所要的注數。但她的記錄已過時。例如:
4 if 6 Howard 要 124 注 但最新紀錄只要 80 就已能保證中 4 個字 (假設你在 21個字內包括 6 個字)
現在還有幾個問題要解決:
1. 最好能選剩多少字才用組合??
想答這問題最好能看機率表，如下:
| Pool | 0 # | 1 # | 2 # | 3# | 4 # | 5 # | 6# |
| 6 | 43.60% | 41.30% | 13.24% | 1.77% | 0.10% | 0.00% | 0.00% |
| 7 | 37.51% | 42.58% | 16.81% | 2.87% | 0.22% | 0.01% | 0.00% |
| 8 | 32.15% | 42.87% | 20.28% | 4.27% | 0.41% | 0.02% | 0.00% |
| 9 | 27.45% | 42.35% | 23.53% | 5.93% | 0.70% | 0.04% | 0.00% |
| 10 | 23.33% | 41.17% | 26.47% | 7.84% | 1.11% | 0.07% | 0.00% |
| 11 | 19.74% | 39.48% | 29.03% | 9.95% | 1.66% | 0.13% | 0.00% |
| 12 | 16.62% | 37.41% | 31.17% | 12.22% | 2.36% | 0.21% | 0.01% |
| 13 | 13.93% | 35.05% | 32.86% | 14.60% | 3.22% | 0.33% | 0.01% |
| 14 | 11.61% | 32.50% | 34.07% | 17.04% | 4.26% | 0.50% | 0.02% |
| 15 | 9.62% | 29.85% | 34.82% | 19.47% | 5.48% | 0.73% | 0.04% |
| 16 | 7.92% | 27.16% | 35.11% | 21.85% | 6.87% | 1.03% | 0.06% |
| 17 | 6.48% | 24.48% | 34.97% | 24.12% | 8.44% | 1.42% | 0.09% |
| 18 | 5.27% | 21.87% | 34.43% | 26.23% | 10.18% | 1.90% | 0.13% |
| 19 | 4.25% | 19.36% | 33.51% | 28.13% | 12.06% | 2.49% | 0.19% |
| 20 | 3.40% | 16.98% | 32.27% | 29.79% | 14.07% | 3.22% | 0.28% |
| 21 | 2.69% | 14.76% | 30.75% | 31.16% | 16.18% | 4.07% | 0.39% |
| 22 | 2.12% | 12.70% | 28.99% | 32.21% | 18.36% | 5.08% | 0.53% |
| 23 | 1.65% | 10.82% | 27.05% | 32.93% | 20.58% | 6.26% | 0.72% |
| 24 | 1.27% | 9.12% | 24.97% | 33.29% | 22.80% | 7.60% | 0.96% |
| 25 | 0.96% | 7.60% | 22.80% | 33.29% | 24.97% | 9.12% | 1.27% |
| 26 | 0.72% | 6.26% | 20.58% | 32.93% | 27.05% | 10.82% | 1.65% |
| 27 | 0.53% | 5.08% | 18.36% | 32.21% | 28.99% | 12.70% | 2.12% |
| 28 | 0.39% | 4.07% | 16.18% | 31.16% | 30.75% | 14.76% | 2.69% |
| 29 | 0.28% | 3.22% | 14.07% | 29.79% | 32.27% | 16.98% | 3.40% |
| 30 | 0.19% | 2.49% | 12.06% | 28.13% | 33.51% | 19.36% | 4.25% |
| 31 | 0.13% | 1.90% | 10.18% | 26.23% | 34.43% | 21.87% | 5.27% |
| 32 | 0.09% | 1.42% | 8.44% | 24.12% | 34.97% | 24.48% | 6.48% |
| 33 | 0.06% | 1.03% | 6.87% | 21.85% | 35.11% | 27.16% | 7.92% |
| 34 | 0.04% | 0.73% | 5.48% | 19.47% | 34.82% | 29.85% | 9.62% |
| 35 | 0.02% | 0.50% | 4.26% | 17.04% | 34.07% | 32.50% | 11.61% |
| 36 | 0.01% | 0.33% | 3.22% | 14.60% | 32.86% | 35.05% | 13.93% |
| 37 | 0.01% | 0.21% | 2.36% | 12.22% | 31.17% | 37.41% | 16.62% |
| 38 | 0.00% | 0.13% | 1.66% | 9.95% | 29.03% | 39.48% | 19.74% |
| 39 | 0.00% | 0.07% | 1.11% | 7.84% | 26.47% | 41.17% | 23.33% |
| 40 | 0.00% | 0.04% | 0.70% | 5.93% | 23.53% | 42.35% | 27.45% |
| 41 | 0.00% | 0.02% | 0.41% | 4.27% | 20.28% | 42.87% | 32.15% |
| 42 | 0.00% | 0.01% | 0.22% | 2.87% | 16.81% | 42.58% | 37.51% |
| 43 | 0.00% | 0.00% | 0.10% | 1.77% | 13.24% | 41.30% | 43.60% |
| 44 | 0% | 0.00% | 0.03% | 0.95% | 9.71% | 38.83% | 50.48% |
| 45 | 0% | 0% | 0.01% | 0.41% | 6.39% | 34.95% | 58.25% |
| 46 | 0% | 0% | 0% | 0.11% | 3.50% | 29.41% | 66.98% |
| 47 | 0% | 0% | 0% | 0% | 1.28% | 21.94% | 76.79% |
| 48 | 0% | 0% | 0% | 0% | 0% | 12.24% | 87.76% |
| 49 | 0% | 0% | 0% | 0% | 0% | 0% | 100% |
例如選 10 球，而你當中能猜中 1 個字的機率是 41.17% 而能選中 6 個字機率機乎是 0. 較合理的大約在 20-25個球。因當中也要考慮成本問題。如上述說的 21 球, 4if6, 80注 成本 $800. 中大獎機會也存在 (參照我之前記錄, 20 個平均能中 4個字).
2. 聰明組合的排列怎排??
聰明組合排列 是一個關於 Optimization 的問題，怎解決?...較出名的就是 Simulated Annealing Algorithm，而近年較新的有 Minimal Overlapping Algorithm. 這些方法目的就是想把會大機會出現的號碼以平均，平衝的狀態放入每注，以確保多幾注能中獎而成本沒有增加。
