Question

A cube is painted on the outside, then cut into 27 equally sized pieces in a 3^3 format. Let X = the total number of painted sides when the 27 pieces are put in a bag and two pieces are randomly selected. Find the probability mass function, the expected value, and the variance of X

Answer 1

below is pmf of Y (painted face on 1 cube):

P(Y=0)=1/27 (since one picece will have 0 painted side)

P(Y=1)=6/27 (since 6 of middle pices on each face will have one painted side)

P(Y=2)=12/27 (4 pieces in each of 3 row)

P(Y=3)=8/27 (a piece in each of 8 corners)

here X =Y1+Y2 (total number of painted face on 2 cube)

below is pmf of X

P(X=0)=P(both contain 0 painted face)=0 (since only one with 0 painted face)

P(X=1)=P(1 with 0 and other with 1 painted face)=2*1*6/(27*26)=2/117

P(X=2)=P(1 with 0 and other wirth 2 painted face)+P(both with 1 painted face)=2*1*12/(27*26)+(6*5)/(27*26)=1/13

P(X=3)=2*1*8/(27*26)+2*6*12/(27*26) =80/351

P(X=4)=2*6*8/(27*26)+12*11/(27*26)=38/117

P(X=5)=2*12*8/(27*26)=32/117

P(X=6)=8*7/(27*26)=28/351

E(X)=0*0+1*2/117+2*1/13+3*80/351+4*38/117+5*32/117+6*28/351 =4

E(X^2)=0^2*0+1^2*2/117+2^2*1/13+3^2*80/351+4^2*38/117+5^2*32/117+6^2*28/351 =17.282

thereofre Var(X)=E(X^2)-E(X)² =1.282