Question

(a) At a carnival stand, they let you roll a die with 1010 sides (labeled 1,2,...,101,2,...,10) and you win that many dimes (for example, rolling a 3 gets you 30 cents). What is the expected dollar value of 88 rolls? (b) On Tuesdays, each roll is squared, and you get that many dimes (e.g. rolling a 3 gets you 90 cents). Now what is the expected dollar value of the 88 rolls?

Answer 1

assuming the die is fair, each side of the die has equal chance of showing up.

let X denotes the value of the die

then P[X=x]=1/1010 where x=1,2,3,....,1010

and let Z denotes the dollar value won.for each roll

then Z=10X/100=X/10

so expected dollar value won in each roll

is E[Z]=E[X/10]=E[X]/10

now E[X]=(1+2+....+1010)/1010=1010*1011/(1010*2)=1011/2

so E[Z}=1011/(2*10)=1011/20=$50.55

hence expected dollar value of 88 rolls is 88*50.55=$4044 [answer]

b) on tuesdays each roll is squared

so then Z=X²/10

so E[Z]=E[X²]/10

now E[X²]=(1*1+2*2+3*3+...+1010*1010)/1010=(1010*1011*(2*1010+1))/(6*1010)=340538.5

so E[Z]=340538.5/10=$34053.85

so expected dollar amount of 88 rolls is 88*34053.85=$2996739 [answer]