Question

A die is rolled 1000 times. The total number of spots is 3680 instead of the expected 3500. Can this be explained as a chance variation, or is the die loaded?

Answer 1

here for a single roll ;

x	P(x)	xP(x)	x2P(x)
1	0.1667	0.167	0.167
2	0.1667	0.333	0.667
3	0.1667	0.500	1.500
4	0.1667	0.667	2.667
5	0.1667	0.833	4.167
6	0.1667	1.000	6.000
	total	3.500	15.167
	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 =	1.708

from above expected value for 1000 rolls=np=1000*3.5 =3500

and standard deviation =1.708*sqrt(1000)=54.0062

as 3680 is more than 3 standard deviation from mean ; therefore this is an unusual event

and we may suspect that the die is loaded.