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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。! M3 O. Z: W, L0 Q0 }& Z
那么，平局、主胜、客胜就有以下表达式：
& f+ F, M6 y- OA(d) = {W(ij)|i=j, i∈N, j∈N}
: n3 z$ z# x. M+ p4 K$ O3 `+ jA(h) = {W(ij)|i>j, i∈N, j∈N}6 m* T; }0 m8 ~+ ^2 W8 s g+ x
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]:0 r' X6 D Z m+ i
P(i)=P(Ai), i∈{0, 1, 2}
& M( F7 }1 X" H, W+ h2 h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）8 M! ~% E7 w1 [" ?6 K
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：
/ _ h0 V' d) y2 H7 i{b(h) , 即上面所述的A(d)结果发生
( V) ^* H \0 G4 j, Z+ SR(1)={O(h)*b(h),即上面所述的A(h)结果发生4 c+ g1 d, q# \, o
{0 , 即上面所述的A(a)结果发生
! u: I( v( w$ } B( ^: n: `# F$ u& j以及
3 [$ { w3 q- g" k! w& ~$ Z{b(a) , 即上面所述的A(d)结果发生- x/ I6 s# s0 g, H. ^3 o! ]* U! T Z
R(2)={0, 即上面所述的A(h)结果发生
9 I) q5 Q3 j# m8 V1 {{O(a)*b(a),即上面所述的A(a)结果发生
" M4 B& f- L4 Z0 m" r7 J- 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5 h6 H% ]" b- S1 S% h
那么投注主客的期望回报总数分别为 m1 ~% X. I8 {. F, @2 j1 C, B4 ]0 ~
E[R(1)]=P(d)*b(h)+P(h)*O(h)*b(h)=b(h)*[P(d)+p(h)O(h)]2 N3 z1 [" `, C' P
E[R(2)]=P(d)*b(a)+P(a)*O(a)*b(a)=b(a)*[P(d)+p(a)O(a)]
) S2 _5 R' E0 o- {$ f* `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
; A; h: ? w9 \E[R(1)]=b(h)*[P(d)+p(h)O(h)]=b(h)
- U' K5 x- u" Q0 A' X+ XP(d)+p(h)O(h)=1! Q* h9 U( S* l9 r& j
O(h)=(1-P(d))/P(h)-----这里就得出平手盘下计算主队赔率的公式
w( J: F+ P! |# C4 ^. g* gE[R(2)]=b(a)*[P(d)+p(a)O(a)]=b(a)
$ P h" A4 g+ V0 n o- @P(d)+p(a)O(a)=18 y! h+ f3 q+ X0 M2 _
O(a)=(1-P(d))/P(a)-----这里就得出平手盘下计算客队赔率的公式6 a1 c; H" C- p
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! v, Q; ]7 u. Y3 Z1 E8 h* ?+ R. S
三、半球盘的计算描述: y' k: q6 e* o% _
接着我们来看看主队（HOME TEAM）受半球（1/2:0 Handicaps的情况）
, H0 v! b; S9 ~$ _9 ^+ F; f. @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：3 A f1 G b3 A, w
{O(h)*b(h),即上面所述的A(d)结果发生
% U& f* z( Q2 w! `# qR(1)={O(h)*b(h),即上面所述的A(h)结果发生* `" X. x( x' N8 Q
{0 , 即上面所述的A(a)结果发生4 r/ Y: l8 I/ |6 Y
以及
0 z6 B6 N: @# h{0 , 即上面所述的A(d)结果发生7 X7 M& Q1 d% n& T7 ^1 o
R(2)={0, 即上面所述的A(h)结果发生* n0 u$ E+ c6 [& W3 _
{O(a)*b(a),即上面所述的A(a)结果发生
* h) A6 `4 G# [8 c B如果O(h)表示受半球盘下的主队赔率，O(a)表示客队赔率/ M: Q8 p6 S& S/ C, c; c" a- u
投注主客的期望回报总数分别为3 m1 x% K, L! R* o4 s. y
E[R(1)]=P(d)*O(h)*b(h)+P(h)*O(h)*b(h)=b(h)*O(h)*[P(d)+p(h)]
" b7 e$ @" `- {% w% X; r' o3 {6 WE[R(2)]=P(a)*O(a)*b(a)
' j1 O) v5 H: ~: X- X2 a7 v9 J& B假设实际投注回报与期望投注回报相等
4 c2 s+ T2 \ {E[R(1)]=b(h)*O(h)*[P(d)+p(h)]=b(h)- G! i4 v; J7 c, w" E2 i
O(h)*[P(d)+p(h)]=1
' j% b$ u5 W7 H1 w! z5 Y1 |O(h)=1/[P(d)+p(h)]9 b0 l3 V) _# L$ W. e
E[R(2)]=P(a)*O(a)*b(a)=b(a): E, A6 s# f1 m$ m! J, @& t! [
P(a)*O(a)=1& v$ `$ p( `9 c5 r" o9 e& t: p8 _4 c
O(a)=1/P(a)
" f( T- @) s! E! e3 l同样道理可以计算让半球（0:1/2 Handicaps）的亚洲盘赔率; e2 o8 R/ K) }; S" }5 z% y
O(h)=1/p(h). ^3 J# ~& S4 P& k4 l' b7 w
O(a)=1/[P(d)+p(a)]9 s, |# p# X0 ^) i2 c; D' C* y. \
四、平半球盘的计算描述
5 \$ }* `2 S; P2 ]+ h* y4 E3 L& [这个稍复杂一点) H z4 t$ ^0 e+ Q! q3 K; I
接着我们来看看主队（HOME TEAM）受平半球（1/4:0 Handicaps的情况）1 I( U: J) G6 h8 F! y* ^( e# I
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：8 v0 [2 v+ V/ S; m9 v C0 Q4 ]
{[O(h)+1]/2×b(h), 即上面所述的A(d)结果发生0 Z7 u1 C$ T$ G
R(1)={O(h)*b(h), 即上面所述的A(h)结果发生7 E0 ]4 Q. M; d8 D, i$ O7 c
{0, 即上面所述的A(a)结果发生. H i" f+ Q: z0 ]. }6 G# K
以及
: \6 P/ S$ ]$ l4 p4 J7 f% I' ~: f{1/2×b(a) , 即上面所述的A(d)结果发生9 e+ B3 X% ~ J# s D0 `4 F
R(2)={0, 即上面所述的A(h)结果发生
: e0 ~% W- G! g{O(a)*b(a), 即上面所述的A(a)结果发生7 ^3 x) L4 @, d5 \4 a
投注主客的期望回报总数分别为 i' r, M* E- Q& n; Y& q
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)）; o% A: k' D9 j) T! T$ M# P
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)]' c! _- I3 }" y4 x
和上面计算过程相似，得出：
# E5 ]6 y8 ~# d+ M7 N% j& T0 I4 BE[R(1)]=b(h)*（P(d)×{[O(h)+1]/2}+P(h)×O(h)）=b(h): U6 \ N9 X2 V9 I' m" {% F% L
P(d)×{[O(h)+1]/2}+P(h)×O(h)=12 M; Y: g9 @9 I% X5 Z+ _
O(h)*[1/2*P(d)+P(h)]+1/2*P(d)=18 y% O5 o7 D9 p o4 D6 |
O(h)=[1-1/2*P(d)]/[1/2*P(d)+P(h)]=（1-P(d)/2）/（P(d)/2+P(h)）
& P) K6 g0 f0 F- l8 _+ VE[R(2)]=b(a)*[1/2×P(d)+P(a)*O(a)]=b(a)
! u. S/ _3 V, k" {) Z, C1 k0 h7 u1/2×P(d)+P(a)*O(a)=1$ v0 ^% [: ^& N
O(a)=[1-1/2×P(d)]/P(a)=（1-P(d)/2）/P(a)8 Z# F$ Z6 A" H0 \% F: P0 K
同样主队让平半就分别是- Z( A u; v1 n% o9 ~6 N* c- N- |
O(h)=（1-P(d)/2）/P(h)
2 l8 [; A: k1 v' c- k& ^O(a)=（1-P(d)/2）/（P(d)/2+P(a)）& y0 H! i# B9 K2 i
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）
# a; b+ `$ q- I3 `, E# C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' N- p3 h& w. {* h1 h' Q) o
B(d) = {W(ij)|i=j+k, i∈N, j∈N,k∈N}
1 X( F% U) _1 T( Q) b1 {B(h) = {W(ij)|i>j+k, i∈N, j∈N,k∈N}, a7 s9 [$ J* @) E4 L3 R4 X v0 s& M
B(a) = {W(ij)|i i,j分别表示主客队的进球数，k代表让球数4 S5 q; _( d) I8 F
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 G& |# w5 L i/ V& k7 `( r{b(h)*O(h) , 即上面所述的B(d)结果发生2 |& k1 ?: c/ F
R(1)={b(h)-O(h)*b(h),即上面所述的B(h)结果发生
# k a7 C* f( U K( T( I) K{0 , 即上面所述的B(a)结果发生) r' p, v; o+ H# T% O) @) j* y8 O. @
以及
2 e0 \7 w5 J, o" Y{b(a) , 即上面所述的B(d)结果发生; g6 x2 U" s& N2 ^' }& b1 t2 r; j5 |3 o
R(2)={0, 即上面所述的B(h)结果发生
1 v) M Q' K& q{O(a)*b(a),即上面所述的B(a)结果发生3 P. ?7 L- x7 u9 ?' E5 c) c
投注主客的期望回报总数分别为
6 c2 C! N" O% ?! P7 W pE[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)]* c7 J4 g1 z9 Y r" i
E[R(2)]=P(h1)*b(a)+（1-P(h)）*O(a)*b(a)=b(a)*（P(h1)+O(a)-P(h)*O(a)）+ O3 e% V3 I! Q: b
假设实际投注回报与期望投注回报相等
- R6 F4 O1 F6 z3 x% w8 I4 ~E[R(1)]=b(h)*[P(h)*O(h)+P(h1)-P(h1)*O(h)]=b(h)$ v) W' o# N) j) J
P(h)*O(h)+P(h1)-P(h1)*O(h)=19 W ^7 s- G- M* ^- ?$ [
O(h)*（P(h)-P(h1)）=1-P(h1)
* \9 i7 i5 d) L# R: f2 RO(h)=（1-P(h1)）/（P(h)-P(h1)）
2 \+ s% z* M, Y! n+ iE[R(2)]=b(a)*（P(h1)+O(a)-P(h)*O(a)）=b(a)2 P, r6 @ z# c6 i- F: `3 S
P(h1)+O(a)-P(h)*O(a)=1
, i% d F7 `& P( m2 U* F9 l8 F7 iO(a)*（1-P(h)）=1-P(h1)
/ D# p" s* O2 n# D/ uO(a)=（1-P(h1)）/（1-P(h)）
- Y! I( W0 Y8 S/ @1 o. [( l8 Y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。3 V, C2 b: h" e G& ~7 x5 o1 c( Q
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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