Question

Problem 6. A fair die is rolled four times. (a) Let Y denote the number of distinct rolls. Find the probability mass function of Y. (b) Let Z denote the minimal result fo the 4 throws. Find the probability mass function of Z

Answer 1

a)total number of outcomes =6⁴ =1296

P(Y=1) =P(all rolls are same)=6/1296 =1/216 (as there are 6 ways all rolls are same)

P(Y=2)=P(select 2 number out of 6 and these two will come at least once in 4 rolls)

=(⁶C₂)*(⁴C₁+⁴C₂+⁴C₃)/1296 =15*(4+6+4)/1296=210/1296=35/216

P(Y=3)=P(select 1 number from 6 and then choose 2 places for it out of 4 and then select 2 number from remaing 5 and put them in remaing two places)=⁶C₁*⁴C₂*⁵C₂*2*1/1296 =720/1296=120/216

P(Y=4)=6*5*4*3/1296=360/1296=60/216

b)

P(Z=1)=P(all number shows 1)=1/1296

P(Z=2)=P(Z<=2)-P(Z=1)=(2⁴/1296)-(1/1296)=15/1296

P(Z=3)=P(Z<=3)-P(Z=2)-P(Z=1)=(3⁴-2⁴-1⁴)/1296=65/1296

P(Z=4)=(4⁴-3⁴-2⁴-1⁴)/1296=175/1296

P(Z=5)=(5⁴-4⁴-3⁴-2⁴-1⁴)/1296=369/1296

P(Z=6)=(6⁴-5⁴-4⁴-3⁴-2⁴-1⁴)/1296=671/1296