Question

Ben climbs the 50 steps to his apartment. At each minute, he rolls a fair die and ascends that number of steps. Assume the rolls are independent. Use Chebechev’s inequality to find a bound on the probability that he will reach his house within 12 minutes.

Answer 1

hre for a die P(x)=1/6 where x=1,2,3,4,5,6

x	f(x)	xP(x)	x2P(x)
1	1/6	0.167	0.167
2	1/6	0.333	0.667
3	1/6	0.500	1.500
4	1/6	0.667	2.667
5	1/6	0.833	4.167
6	1/6	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 =	1.7078

therefore 12 minutes expected steps μ=12*3.5=42

and variance σ² =2.9167*12=35

for Chebychev: P(X-μ>k)>=σ²/(2k²)

hence P(X-42>8) <=35/(2*8²)

P(X>50)<=0.2734