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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。( j; i& Q0 X. f1 K
那么，平局、主胜、客胜就有以下表达式：
+ c% n% W- p* wA(d) = {W(ij)|i=j, i∈N, j∈N}
. C, h" k! Q$ A6 N, r, v- \- A% Y0 c5 q0 xA(h) = {W(ij)|i>j, i∈N, j∈N}
^ I& w4 {9 E( ]8 h: `$ s* I$ h/ k: G! |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]:5 M( a5 u c9 \, |; b; e! S
P(i)=P(Ai), i∈{0, 1, 2}' i6 g3 j, _7 W& l% T
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）0 q, _6 |" ~% P$ l
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：
+ S. X# E4 b/ n J$ u8 \{b(h) , 即上面所述的A(d)结果发生% W9 y% t1 \( `4 _) O# q
R(1)={O(h)*b(h),即上面所述的A(h)结果发生
; f" ]% J3 M. Q" \- C) h) v" a7 l{0 , 即上面所述的A(a)结果发生
: P( r7 Q& z( q以及7 W) j2 a4 _# p8 M# ?+ o& b
{b(a) , 即上面所述的A(d)结果发生
* q Y8 o# v! Z3 i+ `7 H9 _R(2)={0, 即上面所述的A(h)结果发生; o# o- L! T& g
{O(a)*b(a),即上面所述的A(a)结果发生% W5 t Q4 ?. ~+ G1 _0 o
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
/ Z* u: r6 K# z0 s那么投注主客的期望回报总数分别为
3 R, g6 i) \$ E( @$ ?. s% d3 s1 @: j+ pE[R(1)]=P(d)*b(h)+P(h)*O(h)*b(h)=b(h)*[P(d)+p(h)O(h)]
4 p7 x9 j1 m3 T1 h0 ?* eE[R(2)]=P(d)*b(a)+P(a)*O(a)*b(a)=b(a)*[P(d)+p(a)O(a)]8 J( k- B( A7 n: { `* V
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
0 ^& H9 }! J) W- i, g1 @E[R(1)]=b(h)*[P(d)+p(h)O(h)]=b(h)
6 c4 @& ]9 d" b' U' KP(d)+p(h)O(h)=1
) B% T/ u. X0 Q& E; l. [( QO(h)=(1-P(d))/P(h)-----这里就得出平手盘下计算主队赔率的公式) Q5 X* `0 V2 `: V& o: p6 f- ?( f
E[R(2)]=b(a)*[P(d)+p(a)O(a)]=b(a)
1 Q3 }" D9 }$ Y6 y" g- KP(d)+p(a)O(a)=1
2 v. F5 C+ z* d" s, n: zO(a)=(1-P(d))/P(a)-----这里就得出平手盘下计算客队赔率的公式
! D3 ^' ^, H1 e! b: R" _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2 W4 J, {0 W2 n
三、半球盘的计算描述 v9 U. ]+ e9 R+ b, Z) a; S) d5 E7 U
接着我们来看看主队（HOME TEAM）受半球（1/2:0 Handicaps的情况）
$ e+ s4 W4 X7 A+ y; q/ U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：
- h% A8 w- _1 D" D) C! X% ~{O(h)*b(h),即上面所述的A(d)结果发生2 U4 p2 S/ s4 W
R(1)={O(h)*b(h),即上面所述的A(h)结果发生
! |$ Y6 p' v2 E' c1 C- ]2 m{0 , 即上面所述的A(a)结果发生
0 Z5 _2 U0 ~& G以及8 c1 \+ j7 g3 ~, v
{0 , 即上面所述的A(d)结果发生
! u( y* f! Z$ ^# C9 P: {" D# g: `R(2)={0, 即上面所述的A(h)结果发生* R1 R1 g$ I0 @$ F; D, w
{O(a)*b(a),即上面所述的A(a)结果发生
! \0 O. s7 N' D1 R& Y) t; z6 F如果O(h)表示受半球盘下的主队赔率，O(a)表示客队赔率
2 _, w& y# o& W投注主客的期望回报总数分别为6 v: J6 @+ f4 q: V: d- X4 S p/ J8 B
E[R(1)]=P(d)*O(h)*b(h)+P(h)*O(h)*b(h)=b(h)*O(h)*[P(d)+p(h)]0 _) v" M+ V3 _ o& x! B
E[R(2)]=P(a)*O(a)*b(a)
9 c! c6 S$ m+ F, v; M7 T假设实际投注回报与期望投注回报相等
) K# J/ \0 l8 S% t' z& ?E[R(1)]=b(h)*O(h)*[P(d)+p(h)]=b(h)
- k. D0 o% W8 H' _4 N7 P& R5 DO(h)*[P(d)+p(h)]=1
& S3 X, J. B$ F6 u0 oO(h)=1/[P(d)+p(h)]
3 d' p0 U' l, s% g1 F9 M. TE[R(2)]=P(a)*O(a)*b(a)=b(a)
2 w9 w( M8 ?; X6 H4 vP(a)*O(a)=1
5 o' a5 \8 ?3 y( e- Y. \O(a)=1/P(a)
9 e% r+ C% X) d& p+ A* l- T. }$ y同样道理可以计算让半球（0:1/2 Handicaps）的亚洲盘赔率2 A* M4 } T/ K, `+ g" o! r
O(h)=1/p(h)5 V$ y: D0 Q! `: y
O(a)=1/[P(d)+p(a)]
3 [+ M T+ z8 n6 d7 B四、平半球盘的计算描述
/ U* U& Z E: \7 g这个稍复杂一点
; {( o8 s; s+ j- L4 D. ]接着我们来看看主队（HOME TEAM）受平半球（1/4:0 Handicaps的情况）
: J8 r' E7 l, N3 ~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：
$ Z. Z; @9 s4 R! D{[O(h)+1]/2×b(h), 即上面所述的A(d)结果发生 J2 J7 R/ n) u, _! F: D
R(1)={O(h)*b(h), 即上面所述的A(h)结果发生
6 s7 v z& d) {0 l p& K{0, 即上面所述的A(a)结果发生* \7 \# t$ G, A4 |; I K* B
以及* a3 s' k9 @7 H; T" l# \
{1/2×b(a) , 即上面所述的A(d)结果发生
8 w Y- ?( D5 D) K+ Q' D2 F3 H6 A) y- Z& n& IR(2)={0, 即上面所述的A(h)结果发生* y. B. R; { Z% D& [+ Z# T
{O(a)*b(a), 即上面所述的A(a)结果发生% d& `5 `% O4 ~% {# u& J
投注主客的期望回报总数分别为7 ~& h( |7 ~' j& s7 j5 h7 t$ i1 b
E[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)）3 i# d9 R8 M' d! H. E( `
E[R(2)]=1/2×b(a)×P(d)+P(a)*O(a)*b(a)=b(a)*[1/2×P(d)+P(a)*O(a)]" g' B1 R$ o/ c u: u$ W
和上面计算过程相似，得出：9 x$ y4 p/ ` W0 `4 U2 z& @
E[R(1)]=b(h)*（P(d)×{[O(h)+1]/2}+P(h)×O(h)）=b(h)/ Q+ k( d& E& s( O
P(d)×{[O(h)+1]/2}+P(h)×O(h)=18 P, [0 \. B3 S' R% x
O(h)*[1/2*P(d)+P(h)]+1/2*P(d)=1
# G& ^0 Y# Q5 R$ pO(h)=[1-1/2*P(d)]/[1/2*P(d)+P(h)]=（1-P(d)/2）/（P(d)/2+P(h)）3 q6 D/ G* N( \+ R8 T: i
E[R(2)]=b(a)*[1/2×P(d)+P(a)*O(a)]=b(a)
( Q- O' I; N! x2 @& |4 O7 H1 x1/2×P(d)+P(a)*O(a)=10 E2 b. C/ [5 {7 e. P7 e1 i
O(a)=[1-1/2×P(d)]/P(a)=（1-P(d)/2）/P(a)# |! y0 ]5 b/ X' Y$ n. W
同样主队让平半就分别是
4 {' t5 Z1 I( ]$ `O(h)=（1-P(d)/2）/P(h)
2 v9 I2 T- |: b1 Y- PO(a)=（1-P(d)/2）/（P(d)/2+P(a)）+ r* S4 S6 H& w+ J
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）
( j! H; ~7 C1 ^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
' J5 _" i5 ~4 y0 |; WB(d) = {W(ij)|i=j+k, i∈N, j∈N,k∈N}& i+ i: _- H2 |9 m
B(h) = {W(ij)|i>j+k, i∈N, j∈N,k∈N}
1 I! e& u6 j- U5 s4 ?: t4 O+ SB(a) = {W(ij)|i i,j分别表示主客队的进球数，k代表让球数( S, t# W' Z" u) G
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：9 o" Z! k5 @/ U4 R1 i. G8 N
{b(h)*O(h) , 即上面所述的B(d)结果发生" S7 q% ` H \' _/ Q' c1 \0 W
R(1)={b(h)-O(h)*b(h),即上面所述的B(h)结果发生
/ Z4 q( F* }+ {7 I1 g, A) }( Q+ j{0 , 即上面所述的B(a)结果发生
& n) r+ y; Z2 W. u以及
- H) s+ m& S1 f/ {. v" k{b(a) , 即上面所述的B(d)结果发生
: o6 d+ {5 M+ c; }4 [R(2)={0, 即上面所述的B(h)结果发生
& u: A$ s: D/ Q: v; M{O(a)*b(a),即上面所述的B(a)结果发生1 S; \' }9 U* p
投注主客的期望回报总数分别为
/ w( ^* ]7 L; n+ k( [' ]3 ] F1 yE[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)]
5 U- ]' m' i, o9 D0 a6 ZE[R(2)]=P(h1)*b(a)+（1-P(h)）*O(a)*b(a)=b(a)*（P(h1)+O(a)-P(h)*O(a)）
+ C* G* q# Z1 @+ k假设实际投注回报与期望投注回报相等
. Q q0 W2 ]6 _' @E[R(1)]=b(h)*[P(h)*O(h)+P(h1)-P(h1)*O(h)]=b(h)
]1 ^# B; i' e: U8 H. QP(h)*O(h)+P(h1)-P(h1)*O(h)=1
2 G# r. E& I3 k! d, l) o) J9 s5 h% u0 T" IO(h)*（P(h)-P(h1)）=1-P(h1)+ ?, o1 d* D& G2 S6 f
O(h)=（1-P(h1)）/（P(h)-P(h1)）
7 U; X# e8 D$ ^8 B eE[R(2)]=b(a)*（P(h1)+O(a)-P(h)*O(a)）=b(a) _( v4 N. a8 J: R& N
P(h1)+O(a)-P(h)*O(a)=1. r$ ~8 s1 n2 B4 @/ ]( w
O(a)*（1-P(h)）=1-P(h1)
, N. H: G! J& H6 m, O% bO(a)=（1-P(h1)）/（1-P(h)）
9 V2 X" `3 [/ m7 B/ X. u4 Z2 J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。, N+ g) `! l' q- e0 a8 M
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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