Question

1. Game of rolling dice

a. Roll a fair die once. What is the sample space? What is the probability to get “six”? What is the probability to get “six” or “five”?

b. Roll a fair die 10 times. What is the probability to get “six” twice? What is the probability to get six at least twice?

c. Roll a fair die 10 times. What is the expected value and variance of getting “six”?

d. If you roll the die 100 times, how does the probability distribution of getting “six” look like?

Answer 1

(a) sample space of rolling a fair die is S={1,2,3,4,5,6}

the probability to get “six”=P(6)=1/6

the probability to get “six” or “five”=P(6 or 5)=P(6)+P(5)=1/6 + 1/6=2/6=1/3

(b) here we use binomial distribution with parameter n=10, p=1/6 and for

Binomial distribution ,P(X=r)=ⁿC_rp^r(1-p)^n-r  

the probability to get “six” twice=P(X=2)=0.2907 ( using ms-excel =BINOMDIST(2,10,1/6,0))

probability to get six at least twice=P(X>=2)=1-P(X<2)=1-P(X<=1)=0.4845

(using =BINOMDIST(1,10,1/6,1))

(c) expected value getting “six”=n*p=10*(1/6)=5/3=1.667

variance=np(1-p)=10*(1/6)*(1-1/6)=1.3889

(d) probability distribution will be normal distribution as number of rolling is large