强化一件装备，有10%的几率成功，如果未成功，则下次成功几率增加10%（即：第一次未成功，则第二次成功几率为20%），求装备强化成功的期望次数？
function f(deep, p)if p >= 0.99 then -- lua精度问题return deepelseprint("f("..deep..","..p..") = "..deep.."*"..p.." + "..(1-p).." * f("..(deep+1)..","..(p+0.1)..")")return deep*p + (1-p)*f(deep+1,p+0.1)endendprint(f(1, 0.1))-- f(1, 0.1) = 1*0.1 + (1-0.1) * f(2, 0.2)ans = 1*0.1 + 0.9 * (2*0.2 + 0.8 * (3*0.3 + 0.7 * (4*0.4 + 0.6 * (5*0.5 + 0.5 * (6*0.6 + 0.4 * (7*0.7 + 0.3 * (8*0.8 + 0.2 * (9*0.9 + 0.1 * (10*1)))))))))print(ans)-- f(1,0.1) = 1*0.1 + 0.9 * f(2,0.2)-- f(2,0.2) = 2*0.2 + 0.8 * f(3,0.3)-- f(3,0.3) = 3*0.3 + 0.7 * f(4,0.4)-- f(4,0.4) = 4*0.4 + 0.6 * f(5,0.5)-- f(5,0.5) = 5*0.5 + 0.5 * f(6,0.6)-- f(6,0.6) = 6*0.6 + 0.4 * f(7,0.7)-- f(7,0.7) = 7*0.7 + 0.3 * f(8,0.8)-- f(8,0.8) = 8*0.8 + 0.2 * f(9,0.9)-- f(9,0.9) = 9*0.9 + 0.1 * f(10,1)-- f(10,1) = 10*1
略作一般化
某游戏抽卡出货概率是2%，如果连续50次没有出货，接下来每次单抽出货概率+2%，直到出货为止。
即如果一直不出货：
第51次概率为4%
第52次概率为6%
第53次概率为8%
求抽卡出货的期望次数
local add_p = 0.02function f(deep, p)if p >= 0.9999 then -- 精度问题return deepelseif deep<=50 thenprint("f("..deep..","..p..")\t= "..deep.."*"..p.." + "..(1-p).." * f("..(deep+1)..","..(p)..")")return deep*p + (1-p)*f(deep+1,p)elseprint("f("..deep..","..p..")\t= "..deep.."*"..p.." + "..(1-p).." * f("..(deep+1)..","..(p+add_p)..")")return deep*p + (1-p)*f(deep+1,p+add_p)endendendlocal base_p = 0.02print(f(1, base_p))--[[f(1,0.02) = 1*0.02 + 0.98 * f(2,0.02)f(2,0.02) = 2*0.02 + 0.98 * f(3,0.02)f(3,0.02) = 3*0.02 + 0.98 * f(4,0.02)f(4,0.02) = 4*0.02 + 0.98 * f(5,0.02)f(5,0.02) = 5*0.02 + 0.98 * f(6,0.02)f(6,0.02) = 6*0.02 + 0.98 * f(7,0.02)f(7,0.02) = 7*0.02 + 0.98 * f(8,0.02)f(8,0.02) = 8*0.02 + 0.98 * f(9,0.02)f(9,0.02) = 9*0.02 + 0.98 * f(10,0.02)f(10,0.02)= 10*0.02 + 0.98 * f(11,0.02)f(11,0.02)= 11*0.02 + 0.98 * f(12,0.02)f(12,0.02)= 12*0.02 + 0.98 * f(13,0.02)f(13,0.02)= 13*0.02 + 0.98 * f(14,0.02)f(14,0.02)= 14*0.02 + 0.98 * f(15,0.02)f(15,0.02)= 15*0.02 + 0.98 * f(16,0.02)f(16,0.02)= 16*0.02 + 0.98 * f(17,0.02)f(17,0.02)= 17*0.02 + 0.98 * f(18,0.02)f(18,0.02)= 18*0.02 + 0.98 * f(19,0.02)f(19,0.02)= 19*0.02 + 0.98 * f(20,0.02)f(20,0.02)= 20*0.02 + 0.98 * f(21,0.02)f(21,0.02)= 21*0.02 + 0.98 * f(22,0.02)f(22,0.02)= 22*0.02 + 0.98 * f(23,0.02)f(23,0.02)= 23*0.02 + 0.98 * f(24,0.02)f(24,0.02)= 24*0.02 + 0.98 * f(25,0.02)f(25,0.02)= 25*0.02 + 0.98 * f(26,0.02)f(26,0.02)= 26*0.02 + 0.98 * f(27,0.02)f(27,0.02)= 27*0.02 + 0.98 * f(28,0.02)f(28,0.02)= 28*0.02 + 0.98 * f(29,0.02)f(29,0.02)= 29*0.02 + 0.98 * f(30,0.02)f(30,0.02)= 30*0.02 + 0.98 * f(31,0.02)f(31,0.02)= 31*0.02 + 0.98 * f(32,0.02)f(32,0.02)= 32*0.02 + 0.98 * f(33,0.02)f(33,0.02)= 33*0.02 + 0.98 * f(34,0.02)f(34,0.02)= 34*0.02 + 0.98 * f(35,0.02)f(35,0.02)= 35*0.02 + 0.98 * f(36,0.02)f(36,0.02)= 36*0.02 + 0.98 * f(37,0.02)f(37,0.02)= 37*0.02 + 0.98 * f(38,0.02)f(38,0.02)= 38*0.02 + 0.98 * f(39,0.02)f(39,0.02)= 39*0.02 + 0.98 * f(40,0.02)f(40,0.02)= 40*0.02 + 0.98 * f(41,0.02)f(41,0.02)= 41*0.02 + 0.98 * f(42,0.02)f(42,0.02)= 42*0.02 + 0.98 * f(43,0.02)f(43,0.02)= 43*0.02 + 0.98 * f(44,0.02)f(44,0.02)= 44*0.02 + 0.98 * f(45,0.02)f(45,0.02)= 45*0.02 + 0.98 * f(46,0.02)f(46,0.02)= 46*0.02 + 0.98 * f(47,0.02)f(47,0.02)= 47*0.02 + 0.98 * f(48,0.02)f(48,0.02)= 48*0.02 + 0.98 * f(49,0.02)f(49,0.02)= 49*0.02 + 0.98 * f(50,0.02)f(50,0.02)= 50*0.02 + 0.98 * f(51,0.02)f(51,0.02)= 51*0.02 + 0.98 * f(52,0.04)f(52,0.04)= 52*0.04 + 0.96 * f(53,0.06)f(53,0.06)= 53*0.06 + 0.94 * f(54,0.08)f(54,0.08)= 54*0.08 + 0.92 * f(55,0.1)f(55,0.1) = 55*0.1 + 0.9 * f(56,0.12)f(56,0.12)= 56*0.12 + 0.88 * f(57,0.14)f(57,0.14)= 57*0.14 + 0.86 * f(58,0.16)f(58,0.16)= 58*0.16 + 0.84 * f(59,0.18)f(59,0.18)= 59*0.18 + 0.82 * f(60,0.2)f(60,0.2) = 60*0.2 + 0.8 * f(61,0.22)f(61,0.22)= 61*0.22 + 0.78 * f(62,0.24)f(62,0.24)= 62*0.24 + 0.76 * f(63,0.26)f(63,0.26)= 63*0.26 + 0.74 * f(64,0.28)f(64,0.28)= 64*0.28 + 0.72 * f(65,0.3)f(65,0.3) = 65*0.3 + 0.7 * f(66,0.32)f(66,0.32)= 66*0.32 + 0.68 * f(67,0.34)f(67,0.34)= 67*0.34 + 0.66 * f(68,0.36)f(68,0.36)= 68*0.36 + 0.64 * f(69,0.38)f(69,0.38)= 69*0.38 + 0.62 * f(70,0.4)f(70,0.4) = 70*0.4 + 0.6 * f(71,0.42)f(71,0.42)= 71*0.42 + 0.58 * f(72,0.44)f(72,0.44)= 72*0.44 + 0.56 * f(73,0.46)f(73,0.46)= 73*0.46 + 0.54 * f(74,0.48)f(74,0.48)= 74*0.48 + 0.52 * f(75,0.5)f(75,0.5) = 75*0.5 + 0.5 * f(76,0.52)f(76,0.52)= 76*0.52 + 0.48 * f(77,0.54)f(77,0.54)= 77*0.54 + 0.46 * f(78,0.56)f(78,0.56)= 78*0.56 + 0.44 * f(79,0.58)f(79,0.58)= 79*0.58 + 0.42 * f(80,0.6)f(80,0.6) = 80*0.6 + 0.4 * f(81,0.62)f(81,0.62)= 81*0.62 + 0.38 * f(82,0.64)f(82,0.64)= 82*0.64 + 0.36 * f(83,0.66)f(83,0.66)= 83*0.66 + 0.34 * f(84,0.68)f(84,0.68)= 84*0.68 + 0.32 * f(85,0.7)f(85,0.7) = 85*0.7 + 0.3 * f(86,0.72)f(86,0.72)= 86*0.72 + 0.28 * f(87,0.74)f(87,0.74)= 87*0.74 + 0.26 * f(88,0.76)f(88,0.76)= 88*0.76 + 0.24 * f(89,0.78)f(89,0.78)= 89*0.78 + 0.22 * f(90,0.8)f(90,0.8) = 90*0.8 + 0.2 * f(91,0.82)f(91,0.82)= 91*0.82 + 0.18 * f(92,0.84)f(92,0.84)= 92*0.84 + 0.16 * f(93,0.86)f(93,0.86)= 93*0.86 + 0.14 * f(94,0.88)f(94,0.88)= 94*0.88 + 0.12 * f(95,0.9)f(95,0.9) = 95*0.9 + 0.1 * f(96,0.92)f(96,0.92)= 96*0.92 + 0.08 * f(97,0.94)f(97,0.94)= 97*0.94 + 0.059999999999999 * f(98,0.96)f(98,0.96)= 98*0.96 + 0.039999999999999 * f(99,0.98)f(99,0.98)= 99*0.98 + 0.019999999999999 * f(100,1)]]
如果规则再改改，前50次费用固定600，对于第50次后每次抽卡，如果一直不出货，除了概率加2%，费用减6，求出货一次的期望费用。
解法敬请期待…
