Question

Alper, Beatta, and Grandma each pick five cards from a shuffled standard deck. Alper replaces the card and reshuffles each time he picks. Beatta picks from the deck without replacement. Grandma repeatedly picks the top card from the deck and puts it back on the top of the deck. Count an ace as 14, a king as 13, and so on. Let X,Y,Z be the sum of the numbers Alper, Beatta, and Grandma get, respectively. Which of X,Y,Z has or have the largest expected value?
Which of X,Y,Z has or have the largest variance?

Would someone please help me out with correct and detailed answer please?

Answer 1

Only 1 simulation can be skewed so we need to perform it several times to be sure

R code below to copy paste

#Let's perform this experiement once and visualize the outcome
set.seed(100) #so that we can have same random numbers
cards=rep(2:14,4)
cards
#Total cards with points in cards variable as each point is repeated 4 times
alper=sample(size = 5,x = cards,replace = TRUE) # Picks cards with replacement
beatta=sample(size=5,x=cards,replace = FALSE) # Picks cards without replacement
grandma=rep(sample(size=1,x=cards,replace = FALSE),5) #Grandma picks one 1 card 5 times as she puts the same on the deck and repeat to pick 1st

#Now let's simulate the same for 1000 times and find the average of each
  
alper=0
beatta=0
grandma=0
for(i in 1:10000){
cards=rep(2:14,4)
cards
#Total cards with points in cards variable as each point is repeated 4 times
alper=alper+sum(sample(size = 5,x = cards,replace = TRUE)) # Picks cards with replacement
beatta=beatta+sum(sample(size=5,x=cards,replace = FALSE)) # Picks cards without replacement
grandma=grandma+sum(rep(sample(size=1,x=cards,replace = FALSE),5)) #Grandma picks one 1 card 5 times as she puts the same on the deck and repeat to pick 1st
  
}
sum(alper)/10000
sum(beatta)/10000
sum(grandma)/10000

#Hope the above answer has helped you in understanding the problem. Please upvote the ans if it has really #helped you. Good Luck!!