Answer:
R code:
n=1:100*1000
p1=round(dbinom(n,6*n,1/6),4)
p2=round(1-pbinom(n-1,6*n,1/6),4)
levels=factor(c("n","P(A_n)","P(B_n)"))
X=cbind(n,p1,p2)
colnames(X)=c(expression(n),expression(P(A_n)),expression(P(B_n)))
Output:
n P(A_n) P(B_n)
[1,] 1000 0.0138 0.5054
[2,] 2000 0.0098 0.5038
[3,] 3000 0.0080 0.5031
[4,] 4000 0.0069 0.5027
[5,] 5000 0.0062 0.5024
[6,] 6000 0.0056 0.5022
[7,] 7000 0.0052 0.5020
[8,] 8000 0.0049 0.5019
[9,] 9000 0.0046 0.5018
[10,] 10000 0.0044 0.5017
[11,] 11000 0.0042 0.5016
[12,] 12000 0.0040 0.5016
[13,] 13000 0.0038 0.5015
[14,] 14000 0.0037 0.5014
[15,] 15000 0.0036 0.5014
[16,] 16000 0.0035 0.5013
[17,] 17000 0.0034 0.5013
[18,] 18000 0.0033 0.5013
[19,] 19000 0.0032 0.5012
[20,] 20000 0.0031 0.5012
[21,] 21000 0.0030 0.5012
[22,] 22000 0.0029 0.5011
[23,] 23000 0.0029 0.5011
[24,] 24000 0.0028 0.5011
[25,] 25000 0.0028 0.5011
[26,] 26000 0.0027 0.5011
[27,] 27000 0.0027 0.5010
[28,] 28000 0.0026 0.5010
[29,] 29000 0.0026 0.5010
[30,] 30000 0.0025 0.5010
[31,] 31000 0.0025 0.5010
[32,] 32000 0.0024 0.5010
[33,] 33000 0.0024 0.5009
[34,] 34000 0.0024 0.5009
[35,] 35000 0.0023 0.5009
[36,] 36000 0.0023 0.5009
[37,] 37000 0.0023 0.5009
[38,] 38000 0.0022 0.5009
[39,] 39000 0.0022 0.5009
[40,] 40000 0.0022 0.5008
[41,] 41000 0.0022 0.5008
[42,] 42000 0.0021 0.5008
[43,] 43000 0.0021 0.5008
[44,] 44000 0.0021 0.5008
[45,] 45000 0.0021 0.5008
[46,] 46000 0.0020 0.5008
[47,] 47000 0.0020 0.5008
[48,] 48000 0.0020 0.5008
[49,] 49000 0.0020 0.5008
[50,] 50000 0.0020 0.5008
[51,] 51000 0.0019 0.5008
[52,] 52000 0.0019 0.5007
[53,] 53000 0.0019 0.5007
[54,] 54000 0.0019 0.5007
[55,] 55000 0.0019 0.5007
[56,] 56000 0.0018 0.5007
[57,] 57000 0.0018 0.5007
[58,] 58000 0.0018 0.5007
[59,] 59000 0.0018 0.5007
[60,] 60000 0.0018 0.5007
[61,] 61000 0.0018 0.5007
[62,] 62000 0.0018 0.5007
[63,] 63000 0.0017 0.5007
[64,] 64000 0.0017 0.5007
[65,] 65000 0.0017 0.5007
[66,] 66000 0.0017 0.5007
[67,] 67000 0.0017 0.5007
[68,] 68000 0.0017 0.5007
[69,] 69000 0.0017 0.5006
[70,] 70000 0.0017 0.5006
[71,] 71000 0.0016 0.5006
[72,] 72000 0.0016 0.5006
[73,] 73000 0.0016 0.5006
[74,] 74000 0.0016 0.5006
[75,] 75000 0.0016 0.5006
[76,] 76000 0.0016 0.5006
[77,] 77000 0.0016 0.5006
[78,] 78000 0.0016 0.5006
[79,] 79000 0.0016 0.5006
[80,] 80000 0.0015 0.5006
[81,] 81000 0.0015 0.5006
Here we observed that as n goes to infinity, both probabilities are constant and Plan) is almost zero and PB) is almost 0.5.
Explanation:
