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一、我们先假设一个抽样数W={W(ij)|i,j∈N}, W(ij)就是比赛的进球结果，i,j分别表示主客队的进球数。
0 p5 x- G( Q ^- P1 i那么，平局、主胜、客胜就有以下表达式：7 s' a" r: E; M8 q n, F
A(d) = {W(ij)|i=j, i∈N, j∈N}5 |: I+ `) B V' a. J
A(h) = {W(ij)|i>j, i∈N, j∈N}
4 X- e! f. d' ?A(a) = {W(ij)|i 接着我们引入一个符号P(i)表示平胜负的概率，P(Ai)∈[0,1]:
! p8 ]% M6 `' O. C7 r- E* P% a! UP(i)=P(Ai), i∈{0, 1, 2}. B# e; x0 y8 e# } L
二、首先来说说平手盘（即0:0Handicaps或我们经常在国外上看到的(Moneyline）) D* _4 F9 x- {& r
假设b(h)表示主队的投注总数，b(a)表示客队的投注总数，那么投注主客队的回报总数额为：" e1 {( N2 l2 S( u, H- w! ]
{b(h) , 即上面所述的A(d)结果发生
) P% w: f0 s+ G8 t! uR(1)={O(h)*b(h),即上面所述的A(h)结果发生5 e, x) G5 g k9 v2 T% j2 G" `
{0 , 即上面所述的A(a)结果发生+ e! h3 G, M* T
以及) u: x4 g$ W0 U, l2 m3 { m* Z
{b(a) , 即上面所述的A(d)结果发生
, b2 a3 D: S e# r+ f. r6 ^R(2)={0, 即上面所述的A(h)结果发生0 r% }. S2 y4 _0 v$ u1 z6 }
{O(a)*b(a),即上面所述的A(a)结果发生
% R. ?- ^, f: `& r- j8 }, L如果O(h)表示平手盘下的主队赔率，O(a)表示平手盘下的客队赔率
7 L3 ^- ^, g: C% o! h那么投注主客的期望回报总数分别为
! e2 k8 G% O0 JE[R(1)]=P(d)*b(h)+P(h)*O(h)*b(h)=b(h)*[P(d)+p(h)O(h)]
! b; [1 |7 z' {0 c9 xE[R(2)]=P(d)*b(a)+P(a)*O(a)*b(a)=b(a)*[P(d)+p(a)O(a)]
( N6 l4 S$ Y# r% R6 k6 K% l按照真实赔率（暂不包含庄家优势），我们可以认为实际投注回报与期望投注回报相等3 ~4 p7 Z* ~* _* d) ?
E[R(1)]=b(h)*[P(d)+p(h)O(h)]=b(h)
) b1 [5 Y& ?6 V/ }4 R! _ u4 E/ r( LP(d)+p(h)O(h)=1
( c0 n$ o/ t( K/ I4 ]1 a0 T9 gO(h)=(1-P(d))/P(h)-----这里就得出平手盘下计算主队赔率的公式5 X4 a9 b" ^. D4 Z; K
E[R(2)]=b(a)*[P(d)+p(a)O(a)]=b(a)
' A+ z6 K6 g* @" xP(d)+p(a)O(a)=1
0 n1 X, p; I* s8 {7 JO(a)=(1-P(d))/P(a)-----这里就得出平手盘下计算客队赔率的公式
. N6 D5 D& K* q在极端情况下，可以认为平手盘亚洲赔率（或moneyline），就是在不发生平局结果条件下（就是公式中P(d)=0），主胜客胜概率的倒数
# o. ~9 D. c4 d: L; C- N( y: J三、半球盘的计算描述
- D( @4 J1 s; ]: f5 C6 x) r接着我们来看看主队（HOME TEAM）受半球（1/2:0 Handicaps的情况）" P& [2 z% ~1 b9 z/ ?% m) E
还是假设b(h)表示主队的投注总数，b(a)表示客队的投注总数，那么投注主客队的回报总数额为：1 d4 E% t! z' ]! K; @7 S9 J4 Y
{O(h)*b(h),即上面所述的A(d)结果发生" c! Y$ g! o: ^. I" A0 t( r% C
R(1)={O(h)*b(h),即上面所述的A(h)结果发生
% {- M( | b# U! r( W{0 , 即上面所述的A(a)结果发生
* d3 i, ^) l# U# O以及
, L# L2 Y3 D! n/ \; w- L0 N# M+ e; ]{0 , 即上面所述的A(d)结果发生
3 _$ J1 R. M/ a& D, YR(2)={0, 即上面所述的A(h)结果发生1 Z) ?. u5 O' h) ~6 g
{O(a)*b(a),即上面所述的A(a)结果发生1 Z* m! F: i# j9 i" A
如果O(h)表示受半球盘下的主队赔率，O(a)表示客队赔率$ [9 y2 H/ e [# E2 h
投注主客的期望回报总数分别为" h, N: f ?! l* b' s) n
E[R(1)]=P(d)*O(h)*b(h)+P(h)*O(h)*b(h)=b(h)*O(h)*[P(d)+p(h)]0 {/ d' O/ o. L1 j9 t0 _
E[R(2)]=P(a)*O(a)*b(a)9 G& t8 U0 L# f
假设实际投注回报与期望投注回报相等3 Q7 x" h0 X3 c3 v7 g
E[R(1)]=b(h)*O(h)*[P(d)+p(h)]=b(h)4 C, d8 {% f4 f: h
O(h)*[P(d)+p(h)]=1
6 P' f4 r- I" VO(h)=1/[P(d)+p(h)]2 \6 a" @& Q2 `- U
E[R(2)]=P(a)*O(a)*b(a)=b(a)/ i3 m* G: y, z0 q
P(a)*O(a)=1
, v: \( w- ?+ H+ f& @O(a)=1/P(a)& Z! t4 j p& j4 n- y
同样道理可以计算让半球（0:1/2 Handicaps）的亚洲盘赔率
3 a4 c/ m# i! M t$ qO(h)=1/p(h), b* y& w8 Z0 l7 |
O(a)=1/[P(d)+p(a)]
0 q; K% H! N2 X- ?四、平半球盘的计算描述
8 y6 D4 D% V# ]7 S这个稍复杂一点
) |3 u- T, I. C3 @5 x0 K8 S接着我们来看看主队（HOME TEAM）受平半球（1/4:0 Handicaps的情况）
8 N; a) F: D' X4 p) C9 D/ [/ A还是假设b(h)表示主队的投注总数，b(a)表示客队的投注总数，那么投注主客队的回报总数额计算：: y( y. H. r) r Y- H
{[O(h)+1]/2×b(h), 即上面所述的A(d)结果发生$ V; G8 l& |. ^* Q% r% M5 j* B
R(1)={O(h)*b(h), 即上面所述的A(h)结果发生
7 F4 q0 I3 r( S( W p; X! j{0, 即上面所述的A(a)结果发生2 G7 a* D+ C9 {8 {
以及6 x" j7 L, b! e: @, t0 a4 n
{1/2×b(a) , 即上面所述的A(d)结果发生
* O: g( E* W3 \6 ]7 aR(2)={0, 即上面所述的A(h)结果发生
) u9 `* y) o& n' y{O(a)*b(a), 即上面所述的A(a)结果发生
: q8 x; z' P, E$ s$ {5 K- S投注主客的期望回报总数分别为
_6 Y+ P0 ?& _$ G* qE[R(1)]=P(d)×{[O(h)+1]/2}×b(h)+P(h)×O(h)×b(h)=b(h)*（P(d)×{[O(h)+1]/2}+P(h)×O(h)）
+ ^ p: @, b; Z- G; vE[R(2)]=1/2×b(a)×P(d)+P(a)*O(a)*b(a)=b(a)*[1/2×P(d)+P(a)*O(a)]4 | C- K- f3 U) l0 ?" i( p' ]7 A2 Y5 b
和上面计算过程相似，得出：
& Z1 p7 R! x3 M2 f2 _4 G2 fE[R(1)]=b(h)*（P(d)×{[O(h)+1]/2}+P(h)×O(h)）=b(h)+ W* |5 q) p2 i7 L1 P3 ^6 {; U
P(d)×{[O(h)+1]/2}+P(h)×O(h)=1* d/ A1 Z+ z( d0 j7 B, Z9 u
O(h)*[1/2*P(d)+P(h)]+1/2*P(d)=19 W4 i9 r5 F& m g$ j
O(h)=[1-1/2*P(d)]/[1/2*P(d)+P(h)]=（1-P(d)/2）/（P(d)/2+P(h)）! Z3 a1 g( e+ C2 A) m
E[R(2)]=b(a)*[1/2×P(d)+P(a)*O(a)]=b(a)4 z' \. c- B h: W
1/2×P(d)+P(a)*O(a)=1/ m6 i" u7 v) D7 {0 s# m* c, f. n* d0 L
O(a)=[1-1/2×P(d)]/P(a)=（1-P(d)/2）/P(a)
+ g5 G/ h S! Y1 k. w; C; J同样主队让平半就分别是* Q# T v9 w e+ U3 u$ r! o
O(h)=（1-P(d)/2）/P(h)! ^& W4 I* g2 M7 A+ D# L
O(a)=（1-P(d)/2）/（P(d)/2+P(a)）- b' `+ d2 n K' U# I R9 Y
五、一球、两球等整数盘（这里先暂时说主队让1球的情况，0:1，Handicaps，其它可以类推的）
; f9 _% \9 v; S/ s可以采用类似公式，在（一）中我们分别用A(d)、A(h)、A(a)描述平局、主胜、客胜事件的发生。现在改用另外的符号代替，如B(d)、B(h)、B(a)，同时除了平、胜、负概率P(i)（i=d,h,a）外，还需要引入一个一个概率值P(hX)来代表主队赢一球（X=1)、二球(X=2）...的概率，下面来进行演算! R1 K2 h+ n' ?- T7 K8 w# n$ `+ |
B(d) = {W(ij)|i=j+k, i∈N, j∈N,k∈N}
2 `9 s" O4 D0 v' v9 J$ n- {% ^3 KB(h) = {W(ij)|i>j+k, i∈N, j∈N,k∈N}5 }) s0 m @" o& i) f. c# a
B(a) = {W(ij)|i i,j分别表示主客队的进球数，k代表让球数& ^/ g* w5 C% u# z# b
接着，假设b(h)表示主队的投注总数，b(a)表示客队的投注总数，那么投注主客队的回报总数额为：
1 n9 Y* i1 a) z0 Y{b(h)*O(h) , 即上面所述的B(d)结果发生
; J/ A: L* \. PR(1)={b(h)-O(h)*b(h),即上面所述的B(h)结果发生) g& n+ G* C; G
{0 , 即上面所述的B(a)结果发生
9 _" ~$ Z" A( a4 t& n以及 ]% O( F% W- Q& H$ Z% w& `, r
{b(a) , 即上面所述的B(d)结果发生
2 p( f7 S' ~" U/ ?9 Y5 oR(2)={0, 即上面所述的B(h)结果发生- Q# S* {% o3 d; S% ~" c6 x' {" }2 v; K
{O(a)*b(a),即上面所述的B(a)结果发生
# G- k; b5 M, B, _/ E* j& H投注主客的期望回报总数分别为
3 j, b" H6 ?' g/ Z0 P- vE[R(1)]=P(h)*b(h)*O(h)+P(h1)*（b(h)-O(h)*b(h)）=b(h)*[P(h)*O(h)+P(h1)-P(h1)*O(h)]
* A& J1 v2 Q0 I/ `0 ?7 eE[R(2)]=P(h1)*b(a)+（1-P(h)）*O(a)*b(a)=b(a)*（P(h1)+O(a)-P(h)*O(a)）+ s- y4 V" _- c& Z5 P
假设实际投注回报与期望投注回报相等
, d b$ W$ O) g) G: J1 WE[R(1)]=b(h)*[P(h)*O(h)+P(h1)-P(h1)*O(h)]=b(h)! \9 K o- Z* w X
P(h)*O(h)+P(h1)-P(h1)*O(h)=1
' W' k+ {* I3 T4 G( ^O(h)*（P(h)-P(h1)）=1-P(h1)# [$ _0 Y7 M& K# l8 m2 r9 p+ l
O(h)=（1-P(h1)）/（P(h)-P(h1)）
% L9 v0 n# F* A& v! VE[R(2)]=b(a)*（P(h1)+O(a)-P(h)*O(a)）=b(a)
4 A) o( M1 a: P6 F% B$ qP(h1)+O(a)-P(h)*O(a)=1) B9 A" f, a* L+ c, w
O(a)*（1-P(h)）=1-P(h1)# A$ U8 r6 i4 K, n" ~. B
O(a)=（1-P(h1)）/（1-P(h)）
- E4 M( H5 Q z3 s以上（五）部分是让一球的情况，让两球以上整数盘和反过来受让整数盘是可以同样演算的。/ {- x2 [" X2 {. |( _
简单小结一球或整数盘，其实理论的演算过程不难，但是如何准确计算赢整数球的概率（P(hi）就是其中的难点，这已经涉及到如何用相对动态实力差或球差来计算各种赢球概率（是指赢1、2、3...球的概率，也可以说是赢球比分概率）的问题，使用自己数据模型里的数据来计算。 |
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