Question

We are tossing three dice. The final outcome is the sum of the scores. Call it X. What is the Expectation and Variance of X. What is the mode? what is the median of X? What is the 1-standard deviation range? What is the probability that the outcome will be in 1-standard deviation range?

What is the 2-standard deviation range? What is the probability that the outcome will be in the 2-standard deviation range?

Answer 1

here total number of outcomes are 6*6*6=216

below is probaability distribution of sum on 3 dies:

P(X=3)=1/216

P(X=4)=3/216

P(X=5)=6/216

P(X=6)=10/216

P(X=7)=15/216

P(X=8)=21/216

P(X=9)=25/216

P(X=10)=27/216

P(X=11)=27/216

P(X=12)=25/216

P(X=13)=21/216

P(X=14)=15/216

P(X=15)=10/216

P(X=16)=6/216

P(X=17)=3/216

P(X=18)=1/216

from above:

expectation of X =E(X)=10.50

Variance of X =Var(X) =8.75

mode =10,11 (as this have highest probabiliy)

median =10.5 (as 50th percentile or cumulative probability =0.5 falls between 10 and 11)

1 std deviaiton range =mean -/+ std deviation=10.5- /+ 2.958 =7.542 ; 13.458

2-standard deviation range =mean -/+ 2*std deviation=10.5- /+ 2* 2.958 =4.584 ; 16.416

probability that the outcome will be in the 2-standard deviation range =P(5<=X<=16)

=1-(P(X=3)+P(X=4)+P(X=17)+P(X=18)=1-((1/216)+(3/216)+(3/216)+(1/216)=1-(8/216)=208/216=26/27