A fair six-sided die is rolled repeatedly and independently. Let An be the event of rolling n sixes in 6n rolls, and let Bn be the event of rolling n or more sixes in 6n rolls. (a) Does P(An) and P(Bn) change with n? (b) Use a computer to investigate what happens to P(An) and P(Bn) as n becomes very large.

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: