Question

Suppose that we roll a die 208 times. What is the approximate probability that the sum of the numbers obtained is between 685 and 775, inclusive.

Answer 1

for single die:

x	P(X=x)	xP(x)	x2P(x)
1	0.167	0.16667	0.16667
2	0.167	0.33333	0.66667
3	0.167	0.50000	1.50000
4	0.167	0.66667	2.66667
5	0.167	0.83333	4.16667
6	0.167	1.00000	6.00000
total		3.5000	15.1667
	E(x) =μ=	ΣxP(x) =	3.5000
	E(x2) =	Σx2P(x) =	15.1667
	Var(x)=σ2 =	E(x2)-(E(x))2=	2.9167
	std deviation=	σ= √σ2 =	1.7078

expected sum for 208 rolls =208*3.5=728

standard deviation =1.7078*√208 =24.631

Probability that the sum of the numbers obtained is between 685 and 775 :

probability =P(685<X<775)=P((685-728)/24.631)<Z<(775-728)/24.631)=P(-1.75<Z<1.91)=0.9719-0.0401=0.9318

(please try 0.9348 if this comes wrong and reply)