Question

Roll one die 100 times:

What is the probability that the average number face-up is

a) 1

b) 2

c) 3

d) 4

e) 5

g) 6

Could you please calculate the following probabilites and explain how?

Answer 1

her for probability of each outcome on a single roll on a die ; thefore probability of an outcome p=1/6

hence expected value on a single roll E(X)=xP(x) =(1/6)*(1+2+3+4+5+6)=3.5

E(X² )=(1/6)*(1²+2²+3²+4²+5²+6²)=15.1667

std deviation SD(X) =sqrt(E(X² )-(E(X))² )=sqrt(15.1667-3.5²) =1.708

hence for 100 rolls expected mean =3.5

and std deviaiton =1.708/sqrt(100) =0.171

hence from normal dsitribution and continuity correction factor:

a) P(average is 1) =P(X=1) =P(0.5 <X<1.5)=P((0.5-3.5)/0.171<Z<(1.5-3.5)/0.171)=P(-17.54<Z<-11.70)

~0.0000-0.0000 ~ 0.0000

b)

P(X=2) =P(1.5<X<2.5)=P(-11.70<Z<-5.84)~0.0000

c)

P(X=3)=P(2.5<X<3.5)=P(-5.85<Z<0) =0.5-0.0000 =0.5

d)

P(X=4)=P(3.5<X<4.5)=P(0<Z<5.85) =0.5-0.000 =0.5

e)

similarly P(X=5)=0

f) P(X=6) =0